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+{"text":"\\section{Introduction and main results}\n\nUnderstanding the effect of disorder on phase transitions is a key topic in \nstatistical physics. In a celebrated paper, Harris~\\cite{cf:Harris} proposed \na criterion that predicts whether or not the addition of an arbitrarily small \namount of quenched disorder is able to modify the critical behavior of a \nsystem close to a phase transition. The rigorous justification of this criterion \nfor a class of \\emph{pinning models} has been an active direction of research \nin the mathematical literature (see Giacomin~\\cite{cf:Gia2} for an overview). \nOne of the key tools in this program is the \\emph{smoothing inequality} of \nGiacomin and Toninelli~\\cite{cf:GT1}, \\cite{cf:GT2}. It is the purpose of this \nnote to generalize and sharpen this inequality.\n\nSection~\\ref{sec:motivation} provides motivation, Section~\\ref{sec:assumptions}\nstates the necessary model assumptions, Section~\\ref{sec:free energy} defines\nthe free energy, Section~\\ref{sec:theorems} states our main theorems, while \nSection~\\ref{sec:discussion} discusses the context of these theorems. Proofs\nare given in Sections~\\ref{sec:protilt}--\\ref{sec:smoothshift}. \n\n\n\\subsection{Motivation}\n\\label{sec:motivation}\n\nWe begin by describing a class of models that motivates our main results\nin Section~\\ref{sec:theorems}. We use the\nnotation $\\mathbb{N} := \\{1,2,\\ldots\\}$ and $\\mathbb{N}_0 := \\mathbb{N} \\cup \\{0\\}$.\n\nConsider a recurrent Markov chain $S := (S_n)_{n\\in\\mathbb{N}_0}$ on a countable set \n$\\mathsf{E}$, starting at a distinguished point denoted by $0$, defined \non a probability space $(\\Omega,{\\ensuremath{\\mathcal F}} ,\\p)$, and let $\\tau_1 := \\inf\\{n \\in \\mathbb{N}\n\\colon\\, \\ S_n = 0\\}$ be its first return time to $0$. The key assumption is \nthat for some $\\alpha \\in [0,\\infty)$, \n\\begin{equation} \n\\label{eq:astau}\n\\p(\\tau_1 > n) = n^{-\\alpha + o(1)}, \\qquad n \\to \\infty.\n\\end{equation}\nThe case of a transient Markov chain, i.e., $\\p(\\tau_1 = \\infty) > 0$, can be \nincluded as well, and requires that \\eqref{eq:astau} holds conditionally on \n$\\{\\tau_1<\\infty\\}$.\n\nGiven an $\\mathbb{R}$-valued sequence $\\omega := (\\omega_n)_{n\\in\\mathbb{N}}$ (the \n\\emph{disorder} sequence), a function $\\phi\\colon\\,\\mathsf{E} \\to \\mathbb{R}$ \n(the \\emph{potential}), and parameters $N \\in\\mathbb{N}$, $\\beta \\geq 0$, \n$h\\in\\mathbb{R}$ (the \\emph{system size}, the \\emph{disorder strength} and \nthe \\emph{disorder shift}), we define the \\emph{partition function}\n\\begin{equation} \n\\label{eq:model}\nZ_{N, \\beta, h}^{\\omega,\\phi} \\:=\n\\e\\Big[ e^{\\sum_{n=1}^N (h + \\beta\\omega_n) \\phi(S_n)} \\, \n{\\sf 1}_{\\{S_N = 0\\}} \\Big] \\,\\in\\, [0,\\infty],\n\\end{equation}\ni.e., at each time $n$ the Markov chain gets an exponential reward or penalty \nproportional to $h + \\beta \\omega_n$, modulated by a factor $\\phi(S_n)$. The \nsequence $\\omega$ is to be thought of as a typical realization of a random \nprocess. Note that\n\\begin{itemize}\n\\item \nthe choice $\\phi(x) := {\\sf 1}_{\\{0\\}}(x)$ corresponds to the \\emph{pinning model} \n(see Giacomin~\\cite{cf:Gia}, \\cite{cf:Gia2}, den Hollander~\\cite{cf:dH});\n\\item \nwhen $\\mathsf{E} = \\mathbb{Z}$ and $S$ is nearest-neighbor with symmetric excursions\nout of $0$, the choice $\\phi(x) := {\\sf 1}_{(-\\infty,0]}(x)$ corresponds to the \n\\emph{copolymer model} (see \\cite{cf:Gia}, \\cite{cf:dH});\n\\footnote{The standard copolymer model is defined through a \\emph{bond} \ninteraction: $\\phi(S_n)$ is replaced by $\\phi(S_{n-1},S_n) := {\\sf 1}_{(-\\infty, 0]}\n(\\frac{1}{2}[S_{n-1} + S_n])$, and $(\\beta,h)$ by $(-2\\lambda,-2\\lambda h)$. \nThis can be still cast in the framework of \\eqref{eq:model} by picking \n$\\mathsf{E} = \\mathbb{Z}^2$, taking the pair process $(S_{n-1},S_n)$ as the Markov \nchain, and $(0,0)$ as $0$.}\n\\end{itemize}\nThus, the modulating potential $\\phi$ allows us to interpolate between different \nclasses of models. When $S$ is simple random walk on $\\mathbb{Z}^d$ and $\\phi(x) \n\\approx |x|^{-\\theta}$ as $|x| \\to\\infty$ for some $\\theta \\in (0,\\infty)$, the model \ndisplays interesting features that are currently under investigation (Caravenna\nand den Hollander~\\cite{cf:CardH}).\n\n\n\n\\subsection{Assumptions}\n\\label{sec:assumptions}\n\nAlthough our main focus will be on the model in \\eqref{eq:model}, we list the \nassumptions that we actually need. We start with the disorder.\n\n\\begin{assumption}[The disorder]\n\\label{ass:disorder}\nThe disorder $\\omega = (\\omega_n)_{n\\in\\mathbb{N}}$ is an i.i.d.\\ sequence of $\\mathbb{R}$-valued \nrandom variables, defined on a probability space $(\\Omega', {\\ensuremath{\\mathcal F}} ', {\\ensuremath{\\mathbb P}} )$, such that\n\\begin{gather}\n\\label{eq:t0}\n\\exists\\,\\, t_0 \\in (0, \\infty]\\colon\\,\n\\quad \\mathrm{M}(t)  := {\\ensuremath{\\mathbb E}} \\big[ e^{t\\omega_1} \\big] < \\infty \\quad \\forall\\,\\, |t| < t_0, \\\\ \\nonumber\n{\\ensuremath{\\mathbb E}} [\\omega_n] = 0, \\quad \\bbvar(\\omega_n) = 1.\n\\end{gather}\n\\end{assumption}\n\n\\noindent\nThe crucial assumption is that the disorder distribution has locally finite exponential \nmoments. The choice of zero mean and unit variance is a convenient normalization \nonly (since we can play with the parameters $\\beta$ and $h$).\n\nFor $\\delta \\in (-t_0, t_0)$, we denote by ${\\ensuremath{\\mathbb P}} _\\delta$ the \\emph{tilted law} under \nwhich $\\omega = (\\omega_n)_{n\\in\\mathbb{N}}$ is i.i.d.\\ with marginal distribution \n\\begin{equation}\n\\label{eq:RNn}\n{\\ensuremath{\\mathbb P}} _\\delta(\\omega_1 \\in \\mathrm{d} x) := e^{\\delta x - \\log\\mathrm{M}(\\delta)}\\,{\\ensuremath{\\mathbb P}} (\\omega_1 \\in \\mathrm{d} x).\n\\end{equation} \n\nNext we state our assumptions on the partition function $Z_{N,\\omega,\\beta,h}$ we will be\nable to handle, defined for $N \\in \\mathbb{N}$, $\\beta \\geq 0$, $h\\in\\mathbb{R}$ and ${\\ensuremath{\\mathbb P}} $-a.e. $\\omega \\in \n\\mathbb{R}^\\mathbb{N}$ (keeping in mind \\eqref{eq:model} as a special case).\n\n\\begin{assumption}[The partition function {[I]}] \n\\label{ass:model}\n$Z_{N,\\omega,\\beta,h}$ is a\nmeasurable function defined on $\\mathbb{N} \\times \\mathbb{R}^\\mathbb{N} \\times [0,\\infty) \\times \\mathbb{R}$,\ntaking values in $[0,\\infty)$ and satisfying the following conditions:\n\\begin{enumerate}\n\\item\n\\label{it:basic} \n$Z_{N,\\omega,\\beta,h}$ is a function of $N$ and of $(h + \\beta \\omega_n)_{1 \\leq n \\leq N}$.\n\\item\n\\label{it:superadd} \n$Z_{N+M, \\omega, \\beta, h} \\geq Z_{N, \\omega, \\beta, h} \\, Z_{M, \\theta^N\\omega, \\beta, h}$\nfor all $N,M \\in \\mathbb{N}$, where $\\theta$ is the left-shift acting on $\\omega$, i.e., $(\\theta^N \\omega)_n \n:= \\omega_{N+n}$ for $N\\in\\mathbb{N}$.\n\\item\n\\label{it:polylb} \nThere exists a $\\gamma \\in (0,\\infty)$ such that, for $N$ in a subsequence of $\\mathbb{N}$,\n\\begin{equation}\n\\label{eq:polylb}\nZ_{N,\\omega,\\beta,h} \\geq \\frac{c_{\\beta,h}(\\omega)}{N^\\gamma} \n\\quad \\text{with} \\quad\n{\\ensuremath{\\mathbb E}} _\\delta[\\log c_{\\beta,h}(\\omega)] > -\\infty \\quad \\forall\\,\\,\\delta \\in (-t_0, t_0).\n\\end{equation}\n\\end{enumerate}\n\\end{assumption}\n\n\\begin{remark}\\rm \\label{rem:pincop}\nNote that properties \\eqref{it:basic} and \\eqref{it:superadd} are satisfied for the model \nin \\eqref{eq:model}. For property \\eqref{it:polylb} to be satisfied as well, we need \nto make additional assumptions on $\\phi$ and\/or $S$. For instance, for the pinning \nmodel\nproperty \\eqref{it:polylb} holds with $\\gamma = (1+\\alpha)+\\epsilon$, for \nany fixed $\\epsilon > 0$ (and for a suitable choice of $c_{\\beta,h}(\\omega) \n= c^\\epsilon_{\\beta,h}(\\omega)$), which follows from \\eqref{eq:astau} after restricting \nthe expectation in \\eqref{eq:model} to the event $\\{\\tau_1 = N\\}$. Alternatively, when \n$\\mathsf{E} = \\mathbb{Z}^d$, if $\\phi$ vanishes in a half-space and $S$ is symmetric (as for \nthe copolymer model), property \\eqref{eq:polylb} with $\\gamma = (1+\\alpha) +\\epsilon$ \nagain follows from \\eqref{eq:astau}.\n\\end{remark}\n\nAs a matter of fact, properties \\eqref{it:basic} and \\eqref{it:superadd} are rather mild: they \nare satisfied for many $(1+d)$-dimensional directed models (possibly after a minor \nmodification of the partition function that does not change the free energy defined \nbelow). In contrast, property \\eqref{it:polylb} is a more severe restriction. Roughly \nspeaking, it says that the \\emph{disorder can be avoided at a cost that is only \npolynomial in the system size}.\n\n\n\\subsection{Free energy}\n\\label{sec:free energy}\n\nIf Assumptions~\\ref{ass:disorder} and~\\ref{ass:model} are satisfied, then we can \ndefine the \\emph{free energy} \n\\begin{equation} \n\\label{eq:freeen}\n\\textsc{f}(\\beta,h; \\delta) := \\limsup_{N\\to\\infty} \\frac{1}{N} \n{\\ensuremath{\\mathbb E}} _\\delta \\big[ \\log Z_{N, \\omega, \\beta, h} \\big]\n\\end{equation}\nfor $\\beta \\geq 0$, $h \\in \\mathbb{R}$, $\\delta \\in (-t_0, t_0)$ when $\\omega$ is chosen \naccording to ${\\ensuremath{\\mathbb P}} _\\delta$. \n\n\\begin{remark}\\rm\\label{rem:kingman}\n(a) In the general framework of Assumption~\\ref{ass:model}, it may happen that \n$\\textsc{f}(\\beta,h; \\delta) = \\infty$ for some values of the parameters. However, for the \nmodel in \\eqref{eq:model} we have $\\textsc{f}(\\beta,h; \\delta) < \\infty$ as soon as $\\phi$ \nis bounded (see \\eqref{eq:boundham} below).\\\\\n(b) By the super-additivity property \\eqref{it:superadd} in Assumption~\\ref{ass:model}, the $\\limsup$ in \n\\eqref{eq:freeen} may be replaced by $\\sup$, or by $\\lim$ restricted to those values \nof $N$ for which ${\\ensuremath{\\mathbb E}} _\\delta \\big[ \\log Z_{N, \\omega, \\beta, h} \\big] > -\\infty$, which \nby properties \\eqref{it:superadd}--\\eqref{it:polylb} form a sub-lattice $\\textsc{t}\\mathbb{N}$. By \nKingman's super-additive ergodic theorem, we may also remove the expectation \n${\\ensuremath{\\mathbb E}} _\\delta$ in \\eqref{eq:freeen}, because the limit as $N\\to\\infty$, $N \\in \\textsc{t}\\mathbb{N}$, \nexists and is constant ${\\ensuremath{\\mathbb P}} _\\delta$-a.s.\n\\end{remark}\n\nA direct consequence of \\eqref{eq:polylb} is the inequality $\\textsc{f}(\\beta,h; \\delta) \\geq 0$, \nwhich is a crucial feature of the class of models we consider. In many interesting \ncases, like for pinning and copolymer models, the free energy is zero in some \nclosed region of the parameter space and strictly positive in its complement, with\nboth regions non-empty. When this happens, the free energy is not an analytic \nfunction and the model is said to undergo a \\emph{phase transition}. It is then of \nphysical and mathematical interest to study the regularity of the free energy close \nto the \\emph{critical curve} separating the two regions.\n\nMore concretely, consider the case when $h \\mapsto Z_{N,\\omega,\\beta,h}$ is \nmonotone (like for the model in \\eqref{eq:model} when $\\phi$ has a sign), say\nnon-decreasing, so that $h \\mapsto \\textsc{f}(\\beta,h;\\delta)$ is non-decreasing as well. \nThen for every $\\beta \\geq 0$ there exists a critical value $h_c(\\beta) \\in \\mathbb{R} \\cup \n\\{\\pm\\infty\\}$ such that $\\textsc{f}(\\beta,h;0) = 0$ for $h < h_c(\\beta)$ and $\\textsc{f}(\\beta,h;0) \n> 0$ for $h > h_c(\\beta)$ (we consider $\\delta = 0$ for simplicity). If $h \\mapsto \n\\textsc{f}(\\beta,h;0)$ is continuous as well, as is typical, then $\\textsc{f}(\\beta,h_c(\\beta);0) = 0$ \nand it is interesting to understand how the free energy vanishes as $h \\downarrow \nh_c(\\beta)$. For homogeneous pinning models, i.e., when $\\beta = 0$, it is known \nthat\n\\begin{equation} \n\\label{eq:hompin}\n\\textsc{f}(0, h_c(0) + t;0) =  t^{\\max\\{\\frac{1}{\\alpha},1\\} + o(1)}, \\qquad t \\downarrow 0.\n\\end{equation}\n(See \\cite[Theorem~2.1]{cf:Gia} for more precise estimates.) On the other hand,\nas soon as disorder is present, i.e., when $\\beta > 0$, it was shown by Giacomin \nand Toninelli~\\cite{cf:GT1}, \\cite{cf:GT2} that, under some mild restrictions on the \ndisorder distribution,\n\\begin{equation} \n\\label{eq:smoo0}\n\\exists\\,c \\in (0,\\infty)\\colon\\quad 0 \\leq \\textsc{f}(\\beta,h_c(\\beta) + t;0) \n\\leq \\frac{c}{\\beta^2} \\, t^2.\n\\end{equation}\nComparing \\eqref{eq:hompin} and \\eqref{eq:smoo0}, we see that when $\\alpha \n> \\frac{1}{2}$ the addition of disorder has a \\emph{smoothing} effect on the way \nin which the free energy vanishes at the critical line.\n\n\n\\subsection{Main results}\n\\label{sec:theorems}\n\nThe goal of this note is to generalize and sharpen \\eqref{eq:smoo0}, namely, to \nshow that no assumption on the disorder distribution other than \\eqref{eq:t0} is \nrequired, and to provide estimates on the constant $c$ that are optimal in some \nsense (see below). We will stay in the general framework of Assumption~\\ref{ass:model}, \nwith no mention of ``critical lines''. \n\n\n\\subsubsection{$\\bullet$ Tilting}\n\nFirst we prove a smoothing inequality for $\\textsc{f}(\\beta,h;\\delta)$ with respect \nto the tilt parameter $\\delta$ rather than the shift parameter $h$. Although both \ntilting and shifting are natural ways to control the disorder bias, the latter is often \npreferred in the literature because the free energy typically is a convex function of \nthe shift parameter $h$ (like for the model in \\eqref{eq:model}). However, for the \npurpose of the smoothing inequality the tilt parameter $\\delta$ turns out to be more \nnatural.\n\n\\begin{theorem}[Smoothing inequality with respect to a disorder tilt]\n\\label{th:smoothing}\nSubject to Assumptions~{\\rm \\ref{ass:disorder}} and {\\rm \\ref{ass:model}}, if \n$\\textsc{f}(\\bar\\beta, \\bar h; 0) = 0$ for some $\\bar\\beta > 0$ and $\\bar h \\in \\mathbb{R}$, then \nfor all $\\delta \\in (-t_0, t_0)$,\n\\begin{equation} \n\\label{eq:smodelta}\n0 \\leq \\textsc{f}(\\bar\\beta, \\bar h; \\delta) \\leq \\frac{\\gamma}{2} \\, B_\\delta \\, \\delta^2\n\\end{equation}\nwhere the constants $t_0$ and \n$\\gamma$ are defined in \\eqref{eq:t0} and \\eqref{eq:polylb}, while\n\\begin{equation} \nB_\\delta := \\frac{2}{\\delta} \\bigg| (\\log\\mathrm{M})'(\\delta) - \\frac{\\log\\mathrm{M}(\\delta)}{\\delta} \\bigg|\n\\in (0,\\infty)\n\\qquad \\text{satisfies} \\qquad \\lim_{\\delta \\to 0} B_\\delta = 1 .\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\\rm\nFor pinning and copolymer models satisfying \\eqref{eq:astau}, we can set $\\gamma = 1+\\alpha$ \nin \\eqref{eq:smodelta}, by Remark~\\ref{rem:pincop}.\n\\end{remark}\n\n\\noindent\nTheorem~\\ref{th:smoothing} is proved in Section~\\ref{sec:protilt} through a direct translation of \nthe argument developed in Giacomin and Toninelli~\\cite{cf:GT2}. The proof is based on the \nconcept of \\emph{rare stretch strategy}, which has been a crucial tool in the study of disordered \npolymer models since the papers by Monthus~\\cite{cf:Monthus}, Bodineau and \nGiacomin~\\cite{cf:BodGia}.\n\n\n\\subsubsection{$\\bullet$ Shifting}\n\nNext we consider the effect of a disorder shift. In the \\emph{Gaussian case}, i.e., when \n${\\ensuremath{\\mathbb P}} (\\omega_1 \\in \\cdot)=N(0,1)$, tilting is the same as shifting: in fact\n${\\ensuremath{\\mathbb P}} _\\delta (\\omega_1 \\in \n\\cdot)= N(\\delta,1)$ and so $\\omega_n$ under ${\\ensuremath{\\mathbb P}} _\\delta$ is distributed like $\\omega_n \n+ \\delta$ under ${\\ensuremath{\\mathbb P}} $. Recalling property \\eqref{it:basic}, \nwe then get\n\\begin{equation} \n\\label{eq:tiltshift}\n\\textsc{f}(\\beta, h; \\delta) = \\textsc{f}(\\beta, h + \\beta \\delta; 0)\n\\end{equation}\nand, since $\\mathrm{M}(\\delta) = e^{\\delta^2 \/ 2}$, it follows from \\eqref{eq:smodelta} that if $\\textsc{f}(\\bar\\beta, \n\\bar h; 0) = 0$ with $\\bar\\beta > 0$, then\n\\begin{equation}\n0 \\leq \\textsc{f}(\\bar\\beta, \\bar h + t; 0) \\leq \\frac{\\gamma}{2 \\bar\\beta^2} \\, t^2 \n\\qquad \\forall\\, t \\in \\mathbb{R}.\n\\end{equation}\nThis is precisely the smoothing inequality with respect to a disorder shift in \\eqref{eq:smoo0}, \nwith an explicit constant \n(see also Giacomin~\\cite[Theorem 5.6 and Remark 5.7]{cf:Gia}).\n\nFor a general disorder distribution tilting is different from shifting. However, we may still hope \nthat \\eqref{eq:tiltshift} holds approximately. This is what was shown in Giacomin and \nToninelli~\\cite{cf:GT1}, under additional restrictions on the disorder distribution and with \nnon-optimal constants. \nThe main result of this note, Theorem~\\ref{th:compdeltah} below, shows that the effects\non the free energy of tilting or shifting the disorder distribution are asymptotically equivalent, \nin large generality and with asymptotically optimal constants in the weak interaction limit. \nSince this result is unrelated to Theorem~\\ref{th:smoothing} and is of independent interest, \nwe formulate it for a very general class of statistical physics models, way beyond disordered \npolymer models.\n\n\\begin{assumption}[The partition function {[II]}] \n\\label{ass:model2}\nThe partition function is defined as\n\\begin{equation} \n\\label{eq:ZNgen}\nZ_{N, \\omega, \\beta, h} := \\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n}  \\Big],\n\\end{equation}\nwhere, for fixed $N\\in\\mathbb{N}$, $(\\sigma_i)_{1 \\leq i \\leq N}$ are $\\mathbb{R}$-valued measurable functions,\ndefined on a finite measure space $(\\Omega_N, {\\ensuremath{\\mathcal F}} _N, \\p_N)$, that are uniformly bounded,\nhave a sign, say\n\\begin{equation} \n\\label{eq:s0}\n\\exists\\, s_0 > 0\\colon\\, \\quad \\p_N \\big( \\big\\{ 0 \\leq \\sigma_i \\leq s_0, \\, \n\\forall\\, 1 \\leq i \\leq N \\big\\}^c \\big) = 0 \\qquad \\forall\\, N \\in \\mathbb{N},\n\\end{equation}\nand satisfy $-\\infty < \\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(\\Omega_N) < \\infty$.\n\\end{assumption}\n\n\\noindent\nWe emphasize that the $\\sigma_i$'s need not be independent, nor exchangeable.\nA more detailed discussion on Assumption~\\ref{ass:model2} is given below.\n\nWe can now state the approximate version of \\eqref{eq:tiltshift}. The free energy \n$\\textsc{f}(\\beta,h;\\delta)$ is again defined by \\eqref{eq:freeen}.\n\n\\begin{theorem}[Asymptotic equivalence of tilting and shifting]\n\\label{th:compdeltah}\nSubject to Assumptions~{\\rm \\ref{ass:disorder}} and {\\rm \\ref{ass:model2}}, and\nwith $\\epsilon_0 := \\min\\{\\frac{t_0}{2},\\frac{t_0}{2s_0}\\}$ (where $s_0, t_0$ are \ndefined in \\eqref{eq:s0} and \\eqref{eq:t0}), for all $\\beta \\in [0, \\epsilon_0)$ and \n$\\delta \\in (-\\epsilon_0, \\epsilon_0)$ there exist $0 < C^-_{\\beta,\\delta} \\leq \nC^+_{\\beta,\\delta} < \\infty$ such that\n\\begin{equation} \n\\label{eq:compdeltah}\n\\begin{split}\n& \\forall\\, \\delta \\in [0, \\epsilon_0)\\colon\\,\n\\quad \\textsc{f} \\big( \\beta, h + C^{-}_{\\beta,\\delta} \\, \\beta \\delta ; 0 \\big) \\leq\n\\textsc{f} ( \\beta, h; \\delta ) \\leq \\textsc{f} \\big( \\beta, h + C^{+}_{\\beta,\\delta} \\, \\beta \\delta ; 0 \\big),\n\\end{split}\n\\end{equation}\nwhile for $\\delta \\in (-\\epsilon_0, 0]$ the same relation holds with $C^{-}_{\\beta,\\delta}$ \nand $C^{+}_{\\beta,\\delta}$ interchanged. Moreover, $(\\beta,\\delta) \\mapsto \nC^\\pm_{\\beta,\\delta}$ is continuous with $C^\\pm_{0,0} = 1$, and hence\n\\begin{equation}\n\\label{eq:propCpm}\n\\lim_{(\\beta,\\delta) \\to (0,0)} C^\\pm_{\\beta,\\delta} = 1.\n\\end{equation}\nFurthermore, $\\delta \\mapsto C^\\pm_{\\beta,\\delta}\\,\\delta$ is strictly increasing.\n\\end{theorem}\n\n\\noindent\nThe proof of Theorem~\\ref{th:compdeltah} is given in Section~\\ref{sec:tiltshift}. The \ngeneral strategy and consists in showing that the derivatives of $\\textsc{f}(\\beta,h;\\delta)$ \nwith respect to $\\delta$ and $h$ are comparable. Compared to Giacomin and\nToninelli~\\cite{cf:GT1}, several estimates need to be sharpened considerably.\n\n\n\\subsubsection{$\\bullet $ Smoothing}\n\nCombining Theorems~\\ref{th:smoothing} and~\\ref{th:compdeltah}, we finally obtain \nour smoothing inequality with respect to a shift, with explicit control on the constant.\n\n\\begin{theorem}[Smoothing inequality with respect to a disorder shift]\n\\label{th:smoshift}\nSubject to Assumptions~{\\rm \\ref{ass:disorder}}, {\\rm \\ref{ass:model}} and \n{\\rm \\ref{ass:model2}}, there is an $\\epsilon'_0 > 0$ with the following property:\nif $\\textsc{f}(\\bar\\beta, \\bar h; 0) = 0$ for some $\\bar\\beta \\in (0,\\epsilon'_0)$ and \n$\\bar h \\in \\mathbb{R}$, then for $t \\in (-\\bar\\beta \\epsilon'_0, \\bar\\beta \\epsilon'_0)$,\n\\begin{equation} \n\\label{eq:compdeltah2}\n0 \\leq \\textsc{f} ( \\bar\\beta, \\bar h + t; 0) \\leq\n\\frac{\\gamma}{2 \\bar\\beta^2} \\, A_{\\bar\\beta,\\frac{t}{\\bar\\beta}} \\, t^2,\n\\end{equation}\nwhere $(\\beta,\\delta) \\mapsto A_{\\beta,\\delta}$ is continuous from $(0,\\epsilon'_0) \n\\times (\\epsilon'_0, \\epsilon'_0)$ to $(0,\\infty)$, and is such that\n\\begin{equation}\n\t\\lim_{(\\beta,\\delta) \n\t\\to (0,0)} A_{\\beta,\\delta} = 1 .\n\\end{equation}\n\\end{theorem}\n\n\n\\subsection{Discussion}\n\\label{sec:discussion}\n\nWe comment on the results obtained in Section~\\ref{sec:theorems}.\n\n\\medskip\\noindent\n\\textbf{1.}\nThe version of the smoothing inequality in Theorem~\\ref{th:smoshift}, \\emph{with the \nprecision on the constant}, is picked up and used in Berger, Caravenna, Poisat, Sun \nand Zygouras~\\cite{cf:BCPSZ} to obtain the sharp asymptotics of the critical \ncurve $\\beta \\mapsto h_c(\\beta)$ for pinning and copolymer models in the \nweak disorder regime $\\beta \\downarrow 0$, for the case $\\alpha \\in (1,\\infty)$ (recall \n\\eqref{eq:astau}).\n\n\\medskip\\noindent\n\\textbf{2.} \nThe smoothing inequality in \\eqref{eq:compdeltah2},\n\\emph{at the level of generality at which it is stated}, is optimal in the following sense.\n\\begin{itemize}\n\\item \nWe cannot hope for an exponent strictly larger than $2$ in the right-hand side of \n\\eqref{eq:compdeltah2}, because pinning models with $\\P(\\tau_1 = n) \\sim (\\log n)\/n^{3\/2}$ \nare in the ``irrelevant disorder regime'', and it is known that $\\textsc{f}(\\beta,h_c(\\beta)+t; 0) \n\\sim \\textsc{f}(0,h_c(0)+t; 0) = t^{2 + o(1)}$ as $t\\downarrow 0$ for fixed $\\beta > 0$ small \nenough (see Alexander \\cite[Theorem 1.2]{cf:Ken}).\n\\item \nWe cannot hope for an asymptotically smaller constant, i.e., $\\lim_{(\\beta,\\delta) \\to \n(0,0)} A_{\\beta,\\delta} < 1$, because the proof in Berger, Caravenna, Poisat, Sun and\nZygouras~\\cite{cf:BCPSZ} would yield a contradiction (the lower bound would be \nstrictly larger than the upper bound).\n\\end{itemize}\nOf course, for specific models the inequality \\eqref{eq:compdeltah2} can\nsometimes be strengthened.\nFor instance, pinning models satisfying \\eqref{eq:astau} with $\\alpha \\in (0,\\frac{1}{2})$\nare such that $\\textsc{f}(\\beta,h_c(\\beta)+t; 0) \\sim \\textsc{f}(0,h_c(0)+t; 0) = t^{1\/\\alpha + o(1)}$\nas $t \\downarrow 0$ (see \\eqref{eq:hompin}), again by Alexander \\cite[Theorem 1.2]{cf:Ken}.\n\n\\medskip\\noindent\n\\textbf{3.}\nCompared with Assumption~\\ref{ass:model}, Assumption~\\ref{ass:model2} prescribes \na specific form for the partition function $Z_{N, \\omega, \\beta, h}$ and therefore is \nmore restrictive. On the other hand, in view of the minor constraints put on the $\\sigma_i$'s, \n\\eqref{eq:ZNgen} is so general that the absence of any restrictive conditions like \n\\eqref{it:superadd} or \\eqref{it:polylb} makes Assumption~\\ref{ass:model2} effectively \nmuch weaker than Assumption~\\ref{ass:model}. For instance, since \\eqref{eq:model} \nis a special case of \\eqref{eq:ZNgen}, with $\\p_N (\\cdot) = \\p(\\,\\cdot\\, \\cap \\{S_N = 0\\})$\n(which, incidentally, explains why $\\p_N$ is allowed to be a finite measure, and not \nnecessarily a probability), the model in \\eqref{eq:model} satisfies Assumption~\\ref{ass:model2}\nas soon as the function $\\phi$ is bounded and has a sign, without the need for any requirement \nlike \\eqref{eq:astau}. \n\nWe emphasize that many other (also non-directed) disordered models \nfall into Assumption~\\ref{ass:model2}. For instance, for $L \\in\\mathbb{N}$ set $\\Lambda_L := \\{-L, \n\\ldots, +L\\}^d$, $N := |\\Lambda_L| = (2L+1)^d$, $\\Omega_N := \\{-1,+1\\}^{\\Lambda_L}$, \nand let $(\\eta_i)_{i\\in\\Lambda_L}$ be the coordinate projections on $\\Omega_N$. If $\\p_N$ \nis the standard Ising Gibbs measure on $\\Omega_N$, defined by $\\p_N(\\{\\eta_i\\}_{i\\in\\Lambda_L}) \n:= (1\/Z_N) \\exp[ J \\sum_{i,j \\in \\Lambda_L, \\,|i-j| = 1} \\eta_i \\eta_j]$, then the random variables \n$\\sigma_i := \\frac{1}{2}(\\eta_i + 1)$ satisfy Assumption~\\ref{ass:model2}.\n\n\\medskip\\noindent\n\\textbf{4.} \nIt follows easily from \\eqref{eq:freeen} and \\eqref{eq:ZNgen} that (with obvious notation)\n\\begin{equation} \n\\label{eq:compa}\n\\textsc{f}_{(\\sigma_n + c)_{n\\in\\mathbb{N}}} ( \\beta, h; \\delta ) \n= \\textsc{f}_{(\\sigma_n)_{n\\in\\mathbb{N}}} ( \\beta, h; \\delta ) + (\\beta m_\\delta + h) c.\n\\end{equation}\nTherefore, when the $\\sigma_n$'s are uniformly bounded but not necessarily non-negative, we can \nfirst perform a uniform translation to transform them into non-negative random variables, \nnext apply \\eqref{eq:compdeltah}, and finally use \\eqref{eq:compa} to come back to the \noriginal $\\sigma_n$'s. \n\nStill, the non-negativity assumption on the $\\sigma_n$'s in \\eqref{eq:s0}\ncannot be dropped from Theorem~\\ref{th:compdeltah}.\nIn fact, if $\\textsc{f}(\\beta,h;\\delta)$ is differentiable in $h$ and $\\delta$, then\n\\eqref{eq:compdeltah} implies that\n\\begin{equation} \n\\label{eq:tocheck}\n\\forall h \\in \\mathbb{R}\\colon \\qquad\n\\frac{\\partial\\textsc{f}}{\\partial \\delta}(\\beta,h ; 0) \n= \\big[ 1 + o(1) \\big]\\,\\beta \\, \\frac{\\partial\\textsc{f}}{\\partial h}(\\beta,h ; 0),\n\\qquad \\beta \\downarrow 0.\n\\end{equation}\nThis relation, which is a necessary condition for \\eqref{eq:compdeltah} when the free energy is \ndifferentiable, may be violated when the $\\sigma_n$'s take both signs.\nFor instance, let $(\\sigma_n)_{n\\in\\mathbb{N}}$ under $\\p_N := \\p$ be i.i.d.\\ with\n$\\p(\\sigma_n = -1) = \\p(\\sigma_n = +1) = \\frac{1}{2}$, and let the marginal distribution \nof the disorder be ${\\ensuremath{\\mathbb P}} (\\omega_n = -a^{-1}) = a^2\/(a^2+1)$, ${\\ensuremath{\\mathbb P}} (\\omega_n = a) \n= 1\/(a^2+1)$ with $a > 0$ (note that ${\\ensuremath{\\mathbb E}} (\\omega_1) = 0$ and $\\bbvar(\\omega_1) = 1$,\nso that  \\eqref{eq:t0} is satisfied). The free energy is easily computed:\n\\begin{equation}\n\\textsc{f}(\\beta,h ; \\delta) = {\\ensuremath{\\mathbb E}} _\\delta[ \\cosh(h + \\beta \\omega_1) ]\n= \\frac{e^{a\\delta} \\cosh(h + a\\beta) + a^2 e^{-a^{-1}\\delta} \\cosh(h - a^{-1}\\beta)}\n{e^{a\\delta} + a^2 e^{-a^{-1}\\delta}}.\n\\end{equation}\nIn particular,\n\\begin{align}\n\\frac{\\partial\\textsc{f}}{\\partial h} (\\beta,0 ; 0) \n& = \\frac{\\sinh(a\\beta) + a^2 \\sinh(-a^{-1}\\beta)}{1 + a^2} \n= \\frac{a^2 - 1}{6a} \\beta^3  + o(\\beta^3),\\\\\n\\frac{\\partial\\textsc{f}}{\\partial \\delta}(\\beta, 0 ; 0) \n& = \\frac{a \\cosh(a\\beta) - a \\cosh(- a^{-1}\\beta)}{1 + a^2} \n= \\frac{a^2 - 1}{2a} \\, \\beta^2 \\,+\\, o(\\beta^2),\n\\end{align}\nand hence \\eqref{eq:tocheck} does \\emph{not} hold for $a \\ne 1$ (the left-hand side \nis $\\approx \\beta^2$, while the right-hand side is $\\approx \\beta^4$). \nIntuitively, such a discrepancy\narises for values of $h$ at which $\\frac{\\partial\\textsc{f}}{\\partial h}(0,h;0) = 0$, which means that\nthe average $\\e_{N,\\omega, 0,h} (\\frac{1}{N}\\sum_{n=1}^N \\sigma_n)$\ntends to zero as $N\\to\\infty$,\nwhere $\\p_{N,\\omega, \\beta,h}$ is the Gibbs law\nassociated to the partition function $Z_{N, \\omega, \\beta, h}$\n(see \\eqref{eq:PZ} below).\nWhen the $\\sigma_n$'s are non-negative, their individual \nvariances under $\\p_{N,\\omega, 0,h}$ must be small, but this is no longer true when \nthe $\\sigma_n$'s can also take negative values.\nThis is why one might have\n$\\frac{\\partial\\textsc{f}}{\\partial \\delta}(\\beta,h;0) \\gg \\beta\\frac{\\partial\\textsc{f}}{\\partial h}(\\beta,h;0)$\nfor $\\beta > 0$ small\n(compare \\eqref{eq:parth} with \\eqref{eq:fnoy}-\\eqref{eq:pias} below).\n\n\n\n\\smallskip\n\n\\section{Smoothing with respect to a tilt: proof of Theorem~\\ref{th:smoothing}}\n\\label{sec:protilt}\n\n\n\n\\subsection{The $({\\ensuremath{\\mathcal G}} ,{\\ensuremath{\\mathcal C}} )$-rare stretch strategy}\n\\label{sec:rare-stretch-strategy}\n\nFix $\\beta \\geq 0$ and $h \\in \\mathbb{R}$. For $\\ell\\in\\mathbb{N}$, let \n${\\ensuremath{\\mathcal A}} _{\\ell} \\subseteq \\mathbb{R}^{\\ell}$ be a subset of ``disorder stretches'' such that \nthere exist constants ${\\ensuremath{\\mathcal G}}  \\in [0,\\infty)$ and ${\\ensuremath{\\mathcal C}}  \\in [0,\\infty)$ with the following \nproperties, along a diverging sequence of $\\ell \\in \\mathbb{N}$:\n\\begin{itemize}\n\\item \n$\\frac{1}{\\ell} \\log Z_{\\ell,\\omega,\\beta,h} \\geq {\\ensuremath{\\mathcal G}} $ for all $\\omega = \\omega_{(0,\\ell]} \n:= (\\omega_1, \\ldots, \\omega_{\\ell}) \\in {\\ensuremath{\\mathcal A}} _\\ell$ (recall Assumption~\\ref{ass:model}\n\\eqref{it:basic});\n\\item $\\frac{1}{\\ell} \\log {\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell) \\geq - {\\ensuremath{\\mathcal C}} $.\n\\end{itemize}\nThe notation $({\\ensuremath{\\mathcal G}} ,{\\ensuremath{\\mathcal C}} )$ stands for \\emph{gain} versus \\emph{cost}. Recall that \n$\\gamma$ is the exponent in \\eqref{eq:polylb}.\n\n\\begin{lemma}\n\\label{lem:CG}\nThe following implication holds:\n\\begin{equation} \n\\label{eq:loccond}\n{\\ensuremath{\\mathcal G}}  - \\gamma\\, {\\ensuremath{\\mathcal C}}  \\, >  0 \\quad \\Longrightarrow  \\quad \\textsc{f}(\\beta, h ; 0) > 0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFix $\\ell \\in \\mathbb{N}$ large enough so that the above conditions hold, and for $\\omega \\in \\mathbb{R}^\\mathbb{N}$ \ndenote by $T_1(\\omega), T_2(\\omega), \\ldots$ the distances between the endpoints of \nthe stretches in ${\\ensuremath{\\mathcal A}} _\\ell$:\n\\begin{equation}\nT_1(\\omega) := \\inf\\big\\{N \\in \\ell\\mathbb{N}\\colon\\, \\omega_{(N-\\ell,\nN]} \\in {\\ensuremath{\\mathcal A}} _{\\ell}\\big\\},\n\\qquad \nT_{k+1}(\\omega) := T_{1}(\\theta^{T_{1}(\\omega) + \\ldots + T_{k}(\\omega)}(\\omega)).\n\\end{equation}\nNote that $\\{T_k\\}_{k\\in\\mathbb{N}}$ is i.i.d.\\ with marginal law given by $\\ell \\,\\mathrm{GEO}\n({\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _{\\ell}))$. In particular,\n\\begin{equation}\n\t{\\ensuremath{\\mathbb E}} (T_1) = \\ell \/ {\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _{\\ell}) \\le \\ell \\, e^{{\\ensuremath{\\mathcal C}}  \\ell}.\n\\end{equation}\nHenceforth we suppress the subscripts $\\beta,h$. \nSince $(\\theta^{(T_1 + \\ldots + T_{i})-\\ell} \\omega)_{(0,\\ell]} \\in {\\ensuremath{\\mathcal A}} _\\ell$\nby construction, applying\nproperties \\eqref{it:superadd}-\\eqref{it:polylb} in Assumption~\\ref{ass:model}\nand the definition of ${\\ensuremath{\\mathcal G}} $, we get\n\\begin{equation}\nZ_{T_1 + \\ldots + T_k,\\omega} \n\\geq \\prod_{i=1}^k Z_{T_i - \\ell, \\theta^{(T_1 + \\ldots + T_{i-1})} \\omega}\n\\, Z_{\\ell, \\theta^{(T_1 + \\ldots + T_{i})-\\ell} \\omega}\n\\geq e^{k {\\ensuremath{\\mathcal G}}  \\ell} \\, \\prod_{i=1}^k \n\\frac{c_{\\beta,h}(\\theta^{(T_1 + \\ldots + T_{i-1})}\\omega)}{(T_i)^\\gamma},\n\\end{equation}\nwhere we set $Z_0 := 1$ for convenience.\nRecalling \\eqref{eq:freeen} and Remark~\\ref{rem:kingman},\nfor ${\\ensuremath{\\mathbb P}} $-a.e. $\\omega$ we can write, by the strong \nlaw of large numbers and Jensen's inequality,\n\\begin{equation}\n\\begin{split}\n\\textsc{f}(\\beta, h ; 0) \n& \n= \\lim_{k\\to\\infty} \\frac{1}{T_1 + \\ldots + T_k}\n\\log Z_{T_1 + \\ldots + T_k,\\omega,\\beta,h} \\\\\n& \\geq \\frac{1}{{\\ensuremath{\\mathbb E}} (T_1(\\omega))} \\big\\{ \\ell {\\ensuremath{\\mathcal G}}  + {\\ensuremath{\\mathbb E}} [\\log c_{\\beta,h}(\\omega)]\n- \\gamma \\, {\\ensuremath{\\mathbb E}} [\\log (T_1) ] \\big\\} \\\\\n& \\geq \\frac{1}{{\\ensuremath{\\mathbb E}} (T_1(\\omega))} \\big\\{ \\ell {\\ensuremath{\\mathcal G}}  + {\\ensuremath{\\mathbb E}} [\\log c_{\\beta,h}(\\omega)]\n- \\gamma \\, \\log {\\ensuremath{\\mathbb E}} (T_1) \\big\\} \\\\\n& = e^{-{\\ensuremath{\\mathcal C}}  \\ell} \\bigg\\{ ({\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  )\n+ \\frac{{\\ensuremath{\\mathbb E}} [\\log c_{\\beta,h}(\\omega)]}{\\ell} - \\gamma \\frac{\\log \\ell}{\\ell} \\bigg\\}.\n\\end{split}\n\\end{equation}\nIf ${\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  > 0$, then we can choose $\\ell \\in \\mathbb{N}$ large enough (but finite!) \nsuch that the right-hand side is strictly positive. This proves \\eqref{eq:loccond}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{th:smoothing}}\nWe use Lemma~\\ref{lem:CG}. Fix $\\beta > 0$, $h \\in \\mathbb{R}$,\n$\\delta \\in (-t_0, t_0)$ and $\\epsilon > 0$, \nand define the set of good atypical stretches as\n\\begin{equation}\n{\\ensuremath{\\mathcal A}} _{\\ell} := \\bigg\\{ (\\omega_1, \\ldots, \\omega_\\ell) \\in \\mathbb{R}^\\ell\\colon\\, \n\\frac{1}{\\ell}\\log Z_{\\ell, \\omega, \\beta, h} \\geq \\textsc{f}(\\beta, h; \\delta) - \\epsilon \\bigg\\},\n\\end{equation}\nso that ${\\ensuremath{\\mathcal G}}  = \\textsc{f}(\\beta, h; \\delta) - \\epsilon $ by construction. It remains to determine ${\\ensuremath{\\mathcal C}} $, for \nwhich we need to estimate the probability of ${\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell)$ from below. \n\nBy the definition \\eqref{eq:freeen} of $\\textsc{f}(\\beta, h; \\delta)$ together with Kingman's super-additive\nergodic theorem (see Remark~\\ref{rem:kingman}),\nthe event ${\\ensuremath{\\mathcal A}} _\\ell$ is typical for ${ {\\ensuremath{\\mathbb P}} }_\\delta$:\n\\begin{equation} \n\\label{eq:pdelta1}\n\\lim_{\\ell\\to+\\infty} { {\\ensuremath{\\mathbb P}} }_\\delta({\\ensuremath{\\mathcal A}} _\\ell) = 1.\n\\end{equation}\nDenoting by ${\\ensuremath{\\mathbb P}} _\\delta^\\ell$ (resp. ${\\ensuremath{\\mathbb P}} ^\\ell$) the restriction of ${\\ensuremath{\\mathbb P}} _\\delta$\n(resp. ${\\ensuremath{\\mathbb P}} $) on $\\sigma(\\omega_1, \\ldots, \\omega_\\ell)$, we have, by Jensen's inequality\nand \\eqref{eq:RNn},\n\\begin{equation} \\label{eq:entrin}\n\\begin{split}\n\t{\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell) & = {\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\t{\\ensuremath{\\mathbb E}} _\\delta\\bigg( e^{- \\log \\frac{\\mathrm{d} {\\ensuremath{\\mathbb P}} ^\\ell_\\delta}{\\mathrm{d}{\\ensuremath{\\mathbb P}} ^\\ell}}\n\t\\bigg| {\\ensuremath{\\mathcal A}} _\\ell \\bigg) \\ge\n\t{\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\te^{- {\\ensuremath{\\mathbb E}} _\\delta \\big(\\log \\frac{\\mathrm{d} {\\ensuremath{\\mathbb P}} ^\\ell_\\delta}{\\mathrm{d}{\\ensuremath{\\mathbb P}} ^\\ell}\n\t\\big| {\\ensuremath{\\mathcal A}} _\\ell \\big)} \\\\\n\t& = {\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\te^{- \\frac{1}{{\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell)}\n\t{\\ensuremath{\\mathbb E}} _\\delta \\big[ \\big(\n\t\\log \\frac{\\mathrm{d} {\\ensuremath{\\mathbb P}} ^\\ell_\\delta}{\\mathrm{d}{\\ensuremath{\\mathbb P}} ^\\ell}\\big)  \\, {\\sf 1}_{{\\ensuremath{\\mathcal A}} _\\ell} \\big]} \\\\\n\t& =  {\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell) \\,\n\te^{- \\frac{\\ell}{{\\ensuremath{\\mathbb P}} _\\delta({\\ensuremath{\\mathcal A}} _\\ell)}\n\t{\\ensuremath{\\mathbb E}} _\\delta \\big[ \\big(\n\t\\delta \\frac{\\omega_1 + \\ldots + \\omega_\\ell}{\\ell} - \\log \\mathrm{M}(\\delta)\n\t\\big)  \\, {\\sf 1}_{{\\ensuremath{\\mathcal A}} _\\ell} \\big]}\\,.\n\\end{split}\n\\end{equation}\nRecalling \\eqref{eq:RNn} and Assumption~\\ref{ass:disorder}, we abbreviate\n\\begin{equation}\n\\label{eq:mdelta0}\nm_\\delta := {\\ensuremath{\\mathbb E}} _\\delta(\\omega_1) = (\\log\\mathrm{M})'(\\delta)\n= \\delta + o(\\delta), \\qquad \\delta \\to 0.\n\\end{equation}\nBy the strong law of large numbers, it follows from \\eqref{eq:pdelta1}-\\eqref{eq:entrin}\nthat for every $\\epsilon > 0$ we have, for $\\ell$ large enough,\n\\begin{equation}\n\\frac{1}{\\ell} \\log {\\ensuremath{\\mathbb P}} ({\\ensuremath{\\mathcal A}} _\\ell) \\geq \n- \\big[ \\delta \\, m_\\delta - \\log \\mathrm{M}(\\delta) \\big] \\,-\\, \\epsilon =: -{\\ensuremath{\\mathcal C}} ,\n\\end{equation}\n\n\n\n\nWe can conclude.\nWe know from \\eqref{eq:loccond} that $\\textsc{f}(\\beta, h; 0) > 0$ when\n\\begin{equation}\n{\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  = \\textsc{f}(\\beta, h; \\delta) - \\gamma \\big[ \\delta \\, m_\\delta \n- \\log \\mathrm{M}(\\delta) \\big] - 2 \\epsilon > 0.\n\\end{equation}\nIf $\\textsc{f}(\\bar\\beta, \\bar h) = 0$, as in the assumptions of Theorem~\\ref{th:smoothing},\nit follows that ${\\ensuremath{\\mathcal G}}  - \\gamma {\\ensuremath{\\mathcal C}}  \\leq 0$, \ni.e.,\n\\begin{equation}\n\\textsc{f}(\\bar\\beta, \\bar h; \\delta) \\leq \\gamma \n\\big[ \\delta \\, m_\\delta - \\log \\mathrm{M}(\\delta) \\big] + 2 \\epsilon, \\qquad\n\\forall \\delta \\in (-t_0, t_0) \\,.\n\\end{equation}\nSince this equality holds for every $\\epsilon > 0$, it must hold also for $\\epsilon = 0$,\nproving \\eqref{eq:smodelta}.\n\\qed\n\n\n\\smallskip\n\n\\section{Asymptotic equivalence of tilting and shifting: proof of\nTheorem~\\ref{th:compdeltah}}\n\\label{sec:tiltshift}\n\nThroughout this section, we work under Assumptions~{\\rm \\ref{ass:disorder}} \nand {\\rm \\ref{ass:model2}}.\n\n\n\\subsection{Notation}\nDenote the empirical average of the variables $\\sigma_i$'s by\n\\begin{equation} \n\\label{eq:sigmabar}\n\\overline \\sigma_N := \\frac{1}{N} \\sum_{i=1}^N \\sigma_i.\n\\end{equation}\nThe finite-volume Gibbs measure associated with the partition function in \n\\eqref{eq:ZNgen} is the \\emph{probability} on $\\Omega_N$\ndefined, for $N\\in\\mathbb{N}$, $\\omega \\in \\mathbb{R}^\\mathbb{N}$, $\\beta \\geq 0$ \nand $h\\in\\mathbb{R}$, by\n\\begin{equation}\\label{eq:PZ}\n\\begin{split}\n& \\p_{N, \\omega, \\beta, h}(\\,\\cdot\\,) \n:= \\frac{1}{Z_{N, \\omega, \\beta, h}} \n\\e_N \\Big[e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\, {\\sf 1}_{\\{\\cdot\\}} \\Big] \\,, \\\\\n& \\text{where} \\quad\nZ_{N, \\omega, \\beta, h} := \\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\Big]\n\\,. \\end{split}\n\\end{equation}\nLet us spell out the definition \\eqref{eq:freeen} of the free energy, recalling \\eqref{eq:RNn}:\n\\begin{equation} \n\\label{eq:freee}\n\\begin{split}\n\\textsc{f}(\\beta,h; \\delta) \n&:= \\limsup_{N\\to\\infty} \\textsc{f}_N(\\beta,h; \\delta) := \\limsup_{N\\to\\infty} \\frac{1}{N} \n{\\ensuremath{\\mathbb E}} _\\delta\\big[ \\log Z_{N, \\omega, \\beta, h} \\big]  \\\\\n&= \\limsup_{N\\to\\infty} \\frac{1}{N} {\\ensuremath{\\mathbb E}}  \\big[ e^{\\sum_{n=1}^N [\\delta \\omega_n\n- \\log\\mathrm{M}(\\delta)]} \\log Z_{N, \\omega, \\beta, h} \\big].\n\\end{split}\n\\end{equation}\nNote that, by \\eqref{eq:s0},\n\\begin{equation} \n\\label{eq:boundham}\n\\Bigg| \\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n \\Bigg| \n\\leq \\sum_{n=1}^N (|h| + \\beta |\\omega_n|) \\, |\\sigma_n|\n\\leq s_0 \\sum_{n=1}^N (|h| + \\beta |\\omega_n|),\n\\end{equation}\nso that $|\\textsc{f}(\\beta,h;\\delta)| \\leq s_0 (|h| + \\beta {\\ensuremath{\\mathbb E}} _\\delta(|\\omega_1|))\n\\,+\\, |\\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(\\Omega_N)| < \\infty$.\n\n\n\\subsection{Preparation}\n\nBefore proving Theorem~\\ref{th:compdeltah}, we need some preparation. Recalling \n\\eqref{eq:sigmabar}, we define for $[a,b] \\subseteq \\mathbb{R}$ with $a<b$ a restricted version \nof the partition function and the free energy, in which the empirical average\n$\\overline\\sigma_N$ is constrained to lie in $[a,b]$:\n\\begin{equation} \n\\label{eq:freeeab}\n\\begin{split}\nZ_{N, \\omega, \\beta, h}^{[a,b]} \n& := \\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\, \n{\\sf 1}_{\\{\\overline\\sigma_N \\in [a,b]\\}} \\Big],\\\\\n\\textsc{f}^{[a,b]}(\\beta,h; \\delta) \n& := \\limsup_{N\\to\\infty} \\textsc{f}^{[a,b]}_N(\\beta,h; \\delta) \n:= \\limsup_{N\\to\\infty} \\frac{1}{N} {\\ensuremath{\\mathbb E}} _\\delta\\big[ \\log Z_{N, \\omega, \\beta, h}^{[a,b]} \\big].\n\\end{split}\n\\end{equation}\nThe corresponding restricted Gibbs measure is the probability defined by (recall \\eqref{eq:PZ})\n\\begin{equation} \n\\label{eq:pNab}\n\\p_{N, \\omega, \\beta, h}^{[a,b]} (\\,\\cdot\\,) \n:= \\p_{N, \\omega, \\beta, h} (\\,\\cdot\\,|\\,\\overline\\sigma_N \\in [a,b] )\n= \\frac{\\e_N \\Big[ e^{\\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n} \\, \n{\\sf 1}_{\\{\\overline\\sigma_N \\in [a,b]\\}} \\, {\\sf 1}_{\\{\\,\\cdot\\,\\}} \\Big]}{Z_{N, \\omega, \\beta, h}^{[a,b]}}.\n\\end{equation}\n\nNote that $Z_{N, \\omega, \\beta, h} = Z_{N, \\omega, \\beta, h}^{[0,s_0]}$, by \\eqref{eq:s0}.\nFurthermore, $Z_{N, \\omega, \\beta, h}^{[a,b]} \\le Z_{N, \\omega, \\beta, h}^{[c,d]}$ when \n$[a,b] \\subseteq [c,d]$. Therefore\n\\begin{equation} \n\\label{eq:monf}\n\\textsc{f}^{[a,b]}(\\beta,h;\\delta) \\leq \\textsc{f}^{[c,d]}(\\beta,h;\\delta) \\,\\le\\, \\textsc{f} (\\beta,h;\\delta),\n\\qquad [a,b] \\subseteq [c,d].\n\\end{equation}\nIn particular, for $x \\in \\mathbb{R}$ we may define\n\\begin{equation} \n\\label{eq:fx}\n\\textsc{f}^{\\{x\\}}(\\beta,h;\\delta) := \\lim_{n \\to \\infty} \\textsc{f}^{[a_n, b_n]}(\\beta,h;\\delta)\n\\in [-\\infty, +\\infty),\n\\end{equation}\nwhere $a_n \\uparrow x$ and $b_n \\downarrow x$ are arbitrary strictly monotone \nsequences (it is easily seen that the limit does not depend on the choice of these \nsequences). \n\nNote that $\\textsc{f}^{\\{x\\}}(\\beta,h;\\delta) = -\\infty$ when $x \\not\\in [0,s_0]$, \nby \\eqref{eq:s0}. The following result is standard:\n\\begin{equation}\n\\label{eq:fbetah}\n\\textsc{f}(\\beta,h;\\delta) = \\sup_{x \\in [0,s_0]}  \\textsc{f}^{\\{x\\}}(\\beta,h;\\delta).\n\\end{equation}\nIn fact,\nby \\eqref{eq:monf} $\\textsc{f}^{[a,b]}(\\beta,h;\\delta) \\le \\textsc{f}(\\beta,h;\\delta)$ for every $[a,b] \\subseteq \\mathbb{R}$,\nhence by \\eqref{eq:fx} $\\textsc{f}(\\beta,h;\\delta) \\geq \\textsc{f}^{\\{x\\}}(\\beta,h;\\delta)$ for every \n$x\\in\\mathbb{R}$. It follows that the inequality $\\geq$ holds in \\eqref{eq:fbetah}. For the reverse inequality, \nnote that if $a < b < c$, then $[a,c] \\subseteq [a,b] \\cup [b,c]$ and so\n\\begin{equation}\nZ_{N, \\omega, \\beta, h}^{[a,c]} \n\\leq Z_{N, \\omega, \\beta, h}^{[a,b]} + Z_{N, \\omega, \\beta, h}^{[b,c]} \n\\leq 2 \\, \\max \\Big\\{ Z_{N, \\omega, \\beta, h}^{[a,b]},Z_{N, \\omega, \\beta, h}^{[b,c]} \\Big\\}.\n\\end{equation}\nRecalling \\eqref{eq:freeeab}, we see that\n\\begin{equation}\n\\textsc{f}^{[a,c]}(\\beta,h;\\delta) \\leq \n\\max \\Big\\{ \\textsc{f}^{[a,b]}(\\beta,h;\\delta), \\, \\textsc{f}^{[b,c]}(\\beta,h;\\delta) \\Big\\}.\n\\end{equation}\nSince $\\textsc{f}(\\beta,h;\\delta) = \\textsc{f}^{[0,s_0]}(\\beta,h;\\delta)$, we can build a sequence of closed \nintervals $(I_n)_{n\\in\\mathbb{N}_0}$, where $I_0 = [0,s_0]$ and where $I_{n+1}$ is either the first \nhalf or the second half of $I_n$, such that $\\textsc{f}^{I_n}(\\beta,h;\\delta) \\leq \\textsc{f}^{I_{n+1}}(\\beta,h;\\delta)$ \nfor all $n\\in\\mathbb{N}$. In particular,\n\\begin{equation}\n\\textsc{f}(\\beta,h;\\delta) \\leq \\lim_{n\\to\\infty} \\textsc{f}^{I_n}(\\beta,h;\\delta).\n\\end{equation}\nBy compactness, there exists an $\\overline x \\in [0,s_0]$ such that $I_n \\downarrow \n\\{\\overline x\\}$, i.e., $\\bigcap_{n\\in\\mathbb{N}} I_n = \\{\\overline x\\}$. If $I_n = [a_n, b_n]$, then we\nset $J_n := [a_n - \\frac{1}{n}, b_n + \\frac{1}{n}]$, so that we still have $J_n \\downarrow \n\\{\\overline x\\}$, and $\\overline x$ lies in the interior of each $J_n$. Since $\\textsc{f}^{I_n}(\\beta,h;\\delta) \n\\leq \\textsc{f}^{J_n}(\\beta,h;\\delta)$, recalling \\eqref{eq:fx} we obtain\n\\begin{equation}\n\\textsc{f}(\\beta,h;\\delta) \\leq \\lim_{n\\to\\infty} \\textsc{f}^{I_n}(\\beta,h;\\delta)\n\\leq \\lim_{n\\to\\infty} \\textsc{f}^{J_n}(\\beta,h;\\delta) \n= \\textsc{f}^{\\{\\overline x\\}}(\\beta,h;\\delta) \n\\leq \\sup_{x \\in [0,s_0]}  \\textsc{f}^{\\{x\\}}(\\beta,h;\\delta),\n\\end{equation}\nand the proof of \\eqref{eq:fbetah} is complete.\n\n\n\n\\subsection{Proof of Theorem~\\ref{th:compdeltah}}\n\nBy \\eqref{eq:fbetah}, it suffices to show that \\eqref{eq:compdeltah} is satisfied\nwith $\\textsc{f}^{\\{x\\}}$ instead of $\\textsc{f}$, for every fixed $x \\in [0, s_0]$. It is of course \nimportant that the constants $C^\\pm_{\\beta,\\delta}$ do not depend on $x$.\n\n\\medskip\\noindent\n\\textbf{1.}\nFirst we consider the case $x = 0$. We claim that\n\\begin{equation} \n\\label{eq:f0}\n\\textsc{f}^{\\{0\\}}(\\beta,h;\\delta) = \\lim_{\\epsilon \\downarrow 0}\n\\bigg( \\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(0 \\le \\overline\\sigma_N \\le \\epsilon) \\bigg).\n\\end{equation}\nSince the right-hand side of \\eqref{eq:f0} is a constant that does not depend on $\\beta \\ge 0$,\n$\\delta \\in (-t_0,t_0)$ and $h\\in\\mathbb{R}$, \\eqref{eq:compdeltah} is trivially satisfied\nwith $\\textsc{f}^{\\{0\\}}$ instead of $\\textsc{f}$, whatever the \ndefinition of $C^\\pm_{\\beta,\\delta}$ is. To prove \\eqref{eq:f0} note that, by Cauchy-Schwarz,\n\\begin{equation}\n\\begin{split}\n\\Bigg| \\sum_{n=1}^N (h + \\beta \\omega_n) \\sigma_n \\Bigg| \n& \\leq \\sqrt{\\sum_{n=1}^N (h + \\beta \\omega_n)^2} \\, \\sqrt{\\sum_{n=1}^N |\\sigma_n|^2} \n\\leq N \\, s_0 \\, \\sqrt{\\overline\\sigma_N} \\, \\sqrt{\\frac{1}{N} \\sum_{n=1}^N (h + \\beta \\omega_n)^2},\n\\end{split}\n\\end{equation}\nbecause $0 \\leq \\sigma_n = |\\sigma_n| \\leq s_0$ by \\eqref{eq:s0}. Recalling \\eqref{eq:freeeab}, \nfor every $N\\in\\mathbb{N}$ we get\n\\begin{equation}\n\\begin{split}\n\\bigg| \\frac{1}{N} {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\log Z^{[a,b]}_{N,\\omega,\\beta,h} \\big] \n- \\frac{1}{N} \\log \\p_N(a \\le \\overline\\sigma_N \\le b) \\bigg| \n&\\leq  s_0 \\, \\sqrt{b} \\, {\\ensuremath{\\mathbb E}} _\\delta \\left[ \\sqrt{\\frac{1}{N} \n\\sum_{n=1}^N (h + \\beta \\omega_n)^2} \\right] \\\\\n& \\leq s_0 \\, \\sqrt{b} \\,\\sqrt{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (h + \\beta \\omega_1)^2 \\big]},\n\\end{split}\n\\end{equation}\nwhere we use Jensen. Note that the right-hand side is a finite constant. If $|a_N - b_N| \\leq c$ for \nall $N\\in\\mathbb{N}$, then $|\\limsup_N a_N - \\limsup_N b_N| \\leq c$, and so\n\\begin{equation}\n\\bigg| \\textsc{f}^{[a,b]}(\\beta,h;\\delta) \n-  \\bigg(\\limsup_{N\\to\\infty} \\frac{1}{N} \\log \\p_N(a \\le \\overline\\sigma_N \\le b) \\bigg) \\bigg|\n\\leq s_0 \\, \\sqrt{b} \\,\\sqrt{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (h + \\beta \\omega_1)^2 \\big]}.\n\\end{equation}\nTaking $[a,b] = [-\\epsilon, \\epsilon]$ and letting $\\epsilon \\downarrow 0$, we get \\eqref{eq:f0}\nfrom \\eqref{eq:fx} .\n\n\\medskip\\noindent\n\\textbf{2.}\nNext we consider the case $x \\in (0, s_0]$. Roughly speaking, the strategy of the proof is to \nshow that the derivatives of the free energy with respect to $\\delta$ and to $h$ are comparable.\nUnless otherwise specified, we work with generic values of the parameters in the admissible \nrange $\\beta \\geq 0$, $h\\in\\mathbb{R}$ and $\\delta \\in (-t_0, t_0)$. Henceforth we fix $0 < a < b < \\infty$. \nRecalling \\eqref{eq:freeeab} and \\eqref{eq:pNab}, we see that the derivative with respect to $h$ \nof the (restricted) finite-volume free energy $\\textsc{f}^{[a,b]}_N(\\beta, h;\\delta)$ can be expressed as\n\\begin{equation} \n\\label{eq:parth}\n\\frac{\\partial}{\\partial h} \\textsc{f}^{[a,b]}_N(\\beta, h;\\delta) \n= \\frac{1}{N} {\\ensuremath{\\mathbb E}} _\\delta\\bigg[ \\frac{\\partial}{\\partial h} \\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\bigg] \n= {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \\beta, h}\\big[ \\overline\\sigma_N \\big] \\big].\n\\end{equation}\n\n\\medskip\\noindent\n\\textbf{3.}\nThe derivative with respect to $\\delta$ requires some further estimates. Recalling\n\\eqref{eq:PZ}-\\eqref{eq:freee}, we have\n\\begin{equation} \n\\label{eq:start}\n\\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \n= \\frac{1}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_n - m_\\delta) \\,\n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\Big],\n\\end{equation}\nwhere $m_\\delta := {\\ensuremath{\\mathbb E}} _\\delta(\\omega_n) = (\\log \\mathrm{M})'(\\delta)$ by \\eqref{eq:mdelta0}.\nSubtracting a centering term with zero mean, we get\n\\begin{equation} \n\\label{eq:ddf}\n\\begin{split}\n\\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \n& = \\frac{1}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_n - m_\\delta) \\,\n\\Big( \\log Z^{[a,b]}_{N, \\omega, \\beta, h} \n- \\log Z^{[a,b]}_{N, \\omega, \\beta, h}|_{\\omega_n = m_\\delta} \\Big) \\Big] \\\\\n& = \\frac{1}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Bigg[ (\\omega_n - m_\\delta) \n\\int_{m_\\delta}^{\\omega_n} \\bigg( \\frac{\\partial}{\\partial\\omega_n} \n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\bigg) \\bigg|_{\\omega_n = y} \\, \\mathrm{d} y \\Bigg],\n\\end{split}\n\\end{equation}\nwhere we agree that $\\int_a^b (\\ldots) := -\\int_{b}^a(\\ldots)$ when $a > b$.\nAbbreviate\n\\begin{equation} \n\\label{eq:fnoy}\nf_n(\\omega, y) := \\frac{1}{\\beta} \\, \\bigg( \\frac{\\partial}{\\partial \\omega_n} \n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\bigg)\\bigg|_{\\omega_n = y}\n= \\e^{[a,b]}_{N, \\omega, \\beta, h}|_{\\omega_n = y}[\\sigma_n],\n\\end{equation}\nwhere the second equality follows easily from \\eqref{eq:freeeab} via \\eqref{eq:pNab}.\nNote that $f_n(\\omega,y)$ depends on the $\\omega_i$'s for $i \\ne n$, not on \n$\\omega_n$. Therefore \\eqref{eq:ddf} can be rewritten as\n\\begin{equation} \n\\label{eq:pias}\n\\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \n= \\frac{\\beta}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_n - m_\\delta)^2 \\,\n\\frac{1}{\\omega_n - m_\\delta} \\int_{m_\\delta}^{\\omega_n} f_n(\\omega,y) \\, \\mathrm{d} y \\Big].\n\\end{equation}\n\n\\medskip\\noindent\n\\textbf{4.}\nBy \\eqref{eq:fnoy}, the integral average in \\eqref{eq:pias} should be close to\n$\\e^{[a,b]}_{N, \\omega, \\beta, h}[\\sigma_n]$. If we could factorize the expectation\nover ${\\ensuremath{\\mathbb E}} _\\delta$, then the right-hand side in \\eqref{eq:pias} would become \n$\\approx \\beta \\, \\bbvar_\\delta(\\omega_1)\\, {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \n\\beta, h}[\\overline\\sigma_N] \\big]$. Recalling \\eqref{eq:parth}, we see that this is \nprecisely what we want, because $\\bbvar_\\delta(\\omega_1) \\approx 1$ for $\\delta$ \nsmall. In order to turn these arguments into a proof, we need to estimate the dependence \nof $f_n(\\omega, y)$ on $y$. To that end we note that\n\\begin{equation}\n\\label{eq:varus2}\n\\begin{split}\n\\frac{\\partial}{\\partial \\omega_n} f_n(\\omega, \\omega_n) \n& = \\frac{1}{\\beta} \\frac{\\partial^2}{\\partial \\omega_n^2}\n\\log Z^{[a,b]}_{N, \\omega, \\beta, h} \\,=\\, \\beta \\, \n\\var_{N, \\omega, \\beta, h}^{[a,b]}[\\sigma_n] \\\\\n& \\leq \\beta \\, \\e^{[a,b]}_{N, \\omega, \\beta, h} [\\sigma_n^2] \n\\leq  s_0 \\, \\beta \\, \\e^{[a,b]}_{N, \\omega, \\beta, h}[\\sigma_n] \n= s_0 \\, \\beta \\,  f_n(\\omega, \\omega_n)\n\\end{split}\n\\end{equation}\nbecause $0 \\le \\sigma_n \\le s_0$, by \\eqref{eq:s0}. Therefore\n\\begin{equation}\n\\frac{\\partial}{\\partial y} f_n(\\omega, y) \\geq 0, \n\\qquad \\frac{\\partial}{\\partial y} \\big( e^{-s_0 \\, \\beta \\, y} \\, f_n(\\omega, y) \\big) \\leq\t0,\n\\end{equation}\nand integrating these relations we get\n\\begin{equation} \n\\label{eq:ustec}\ne^{-s_0 \\beta  (y - y')^-} \\, f_n(\\omega, y') \n\\leq f_n(\\omega, y) \\leq e^{s_0 \\beta  (y - y')^+} \\, f_n(\\omega, y') \n\\qquad \\forall\\, y,y' \\in \\mathbb{R}.\n\\end{equation}\nIntroducing the function\n\\begin{equation} \n\\label{eq:defg}\ng(x) := \\begin{cases}\n\\displaystyle\n\\frac{e^{x} - 1}{x} \n& \\text{if } x \\ne 0, \\\\\n1 \n& \\text{if } x = 0,\n\\end{cases}\n\\end{equation}\ntaking $y' = m_\\delta$ in \\eqref{eq:ustec} and integrating over $y$, we easily obtain the bounds\n\\begin{equation} \n\\label{eq:g-+est}\n\\begin{split}\ng\\big(-\\beta s_0(\\omega_n - \n& m_\\delta)^-\\big) \\, f_n(\\omega, m_\\delta) \\\\\n& \\leq \\frac{1}{\\omega_n - m_\\delta}\n\\int_{m_\\delta}^{\\omega_n} f_n(\\omega,y) \\, \\mathrm{d} y \n\\leq g\\big(\\beta s_0(\\omega_n - m_\\delta)^+\\big) \\, f_n(\\omega, m_\\delta).\n\\end{split}\n\\end{equation}\n\n\\medskip\\noindent\n\\textbf{5.}\nBefore inserting this estimate into \\eqref{eq:pias}, let us pause for a brief integrability interlude.\nThe random variable $g(-\\beta s_0(\\omega_n - m_\\delta)^-)$ is bounded, so there is no integrability \nconcern. On the other hand, the random variable $g(\\beta s_0(\\omega_n - m_\\delta)^+)$ is \nunbounded and a little care is required. Note that\n\\begin{equation}\ng(\\beta s_0(\\omega_n - m_\\delta)^+) \\leq A + B \\, e^{\\beta s_0 \\omega_n }\n\\end{equation}\nfor $A,B>0$, and that ${\\ensuremath{\\mathbb E}} _\\delta(e^{t\\omega_1}) < \\infty$ for $t + \\delta \\in (-t_0, +t_0)$, by \n\\eqref{eq:t0} and \\eqref{eq:RNn}. Therefore, when we integrate  $g(\\beta s_0(\\omega_n - \nm_\\delta)^+)$ (possibly times a polynomial of $\\omega_n$) over ${\\ensuremath{\\mathbb P}} _\\delta$, to have a \nfinite outcome we need to ensure that $\\beta s_0 + \\delta \\in (-t_0, +t_0)$. This is simply \nachieved through the restrictions $\\delta \\in (-\\epsilon_0 ,\\epsilon_0)$ and $\\beta \\in [0,\\epsilon_0)$,\nwhere $\\epsilon_0 := \\min\\{\\frac{t_0}{2},\\frac{t_0}{2s_0}\\}$,\nas in the statement of Theorem~\\ref{th:compdeltah}.\nWe make these restrictions henceforth.\n\n\\medskip\\noindent\n\\textbf{6.}\nLet us now substitute the estimate \\eqref{eq:g-+est} into \\eqref{eq:pias}. Since $f_n(\\omega, \nm_\\delta)$ does not depend on $\\omega_n$, the expectation over ${\\ensuremath{\\mathbb E}} _\\delta$ factorizes \nand we obtain\n\\begin{equation} \n\\label{eq:aatt}\n\\begin{split}\n{\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_1 - m_\\delta)^2 \\, \n& g\\big(-\\beta s_0 (\\omega_1 - m_\\delta)^-\\big) \\Big]\n\\, \\Bigg( \\frac{\\beta}{N} \\sum_{n=1}^N\n{\\ensuremath{\\mathbb E}} _\\delta \\Big[ f_n(\\omega, m_\\delta) \\Big] \\Bigg) \\\\\n& \\leq \\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta) \\\\\n& \\leq {\\ensuremath{\\mathbb E}} _\\delta \\Big[ (\\omega_1 - m_\\delta)^2 \\, g\\big(\\beta s_0 (\\omega_1 - m_\\delta)^+\\big) \\Big]\n\\, \\Bigg( \\frac{\\beta}{N} \\sum_{n=1}^N {\\ensuremath{\\mathbb E}} _\\delta \\Big[ f_n(\\omega, m_\\delta) \\Big] \\Bigg).\n\\end{split}\n\\end{equation}\nWe next want to replace $f_n(\\omega, m_\\delta)$ by $f_n(\\omega, \\omega_n) \n= \\e^{[a,b]}_{N, \\omega, \\beta, h} \\big[ \\sigma_n \\big]$ (recall \\eqref{eq:fnoy}). \nTo this end, we again apply \\eqref{eq:ustec}, this time with $y = \\omega_n$ and \n$y' = m_\\delta$. Since $f_n(\\omega, m_\\delta)$ does not depend on $\\omega_n$, \nwe have\n\\begin{equation} \n\\label{eq:aatt+}\n\\begin{split}\n\\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ f_n(\\omega, \\omega_n) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[e^{s_0 \\beta  (\\omega_1 - m_\\delta)^+ } \\big]}\n\\leq {\\ensuremath{\\mathbb E}} _\\delta \\Big[ f_n(\\omega, m_\\delta) \\Big] \n& \\leq \\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ f_n(\\omega, \\omega_n) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[ e^{-s_0 \\beta (\\omega_1 - m_\\delta)^-} \\big]}.\n\\end{split}\n\\end{equation}\nWe can now introduce the constants\n\\begin{equation} \n\\label{eq:c+-}\n\\begin{split}\nc^+_{\\beta, \\delta} \n& := \\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (\\omega_1 - m_\\delta)^2 \n\\, g\\big(\\beta s_0 (\\omega_1 - m_\\delta)^+\\big) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[ e^{-s_0 \\beta (\\omega_1 - m_\\delta)^-} \\big]}, \\\\\nc^-_{\\beta, \\delta} \n& := \\frac{{\\ensuremath{\\mathbb E}} _\\delta \\big[ (\\omega_1 - m_\\delta)^2 \n\\, g\\big(-\\beta s_0 (\\omega_1 - m_\\delta)^-\\big) \\big]}\n{{\\ensuremath{\\mathbb E}} _\\delta \\big[ e^{s_0 \\beta  (\\omega_1 - m_\\delta)^+ } \\big]},\n\\end{split}\n\\end{equation}\nand note that $0 < c^-_{\\beta, \\delta} \\le c^+_{\\beta, \\delta} < \\infty$ for all $\\delta \n\\in (-\\epsilon_0, \\epsilon_0)$ and $\\beta \\in [0,\\epsilon_0)$. We have already observed \nthat $f_n(\\omega, \\omega_n) = \\e^{[a,b]}_{N, \\omega, \\beta, h} \\big[ \\sigma_n \\big]$\nby \\eqref{eq:fnoy}, and so from \\eqref{eq:aatt}-\\eqref{eq:aatt+} we obtain the following \nestimate: for every $\\beta \\in [0,\\epsilon_0)$, $h\\in\\mathbb{R}$, $\\delta \\in (-\\epsilon_0, \\epsilon_0)$ \nand $0 < a < b <\\infty$\n\\begin{equation} \n\\label{eq:partdelta}\nc^-_{\\beta,\\delta} \\, \\beta \\, {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \\beta, h}\n\\big[ \\overline\\sigma_N \\big] \\big] \n\\leq \\frac{\\partial}{\\partial\\delta} \\textsc{f}^{[a,b]}_N(\\beta, h; \\delta)\n\\leq c^+_{\\beta,\\delta} \\, \\beta \\, {\\ensuremath{\\mathbb E}} _\\delta \\big[ \\e^{[a,b]}_{N, \\omega, \\beta, h}\n\\big[ \\overline\\sigma_N \\big] \\big].\n\\end{equation}\nNote the analogy with the expression in \\eqref{eq:parth} for $\\frac{\\partial}{\\partial h}\n\\textsc{f}^{[a,b]}_N(\\beta, h; \\delta)$.\n\n\\medskip\\noindent\n\\textbf{7.}\nWe are close to the final conclusion. Since by \\eqref{eq:pNab} we have $a \\leq \n\\e^{[a,b]}_{N, \\omega, \\beta, h}\\big[ \\overline\\sigma_N \\big] \\leq b$, it follows from \n\\eqref{eq:partdelta} that, for every $\\delta \\in [0,\\epsilon_0)$\n\\begin{equation} \n\\label{eq:cru0}\nC^-_{\\beta,\\delta} \\, \\beta \\, a \\, \\delta \n\\leq \\textsc{f}^{[a,b]}(\\beta, h; \\delta) - \\textsc{f}^{[a,b]}(\\beta, h; 0) \n\\leq C^+_{\\beta,\\delta} \\, \\beta \\, b \\, \\delta,\n\\end{equation}\nwhere we set\n\\begin{equation}\n\\label{eq:C+-}\nC^\\pm_{\\beta,\\delta} := \n\\begin{cases}\n\\displaystyle \\frac{1}{\\delta} \\int_0^\\delta \nc^\\pm_{\\beta,\\delta'} \\, \\mathrm{d} \\delta' \n& \\text{if } \\delta \\in (-\\epsilon_0, \\epsilon_0) \\setminus\\{0\\}, \\\\\nc^\\pm_{\\beta,0} \n& \\text{if } \\delta = 0.\n\\end{cases}\n\\end{equation}\nAnalogously to \\eqref{eq:cru0}, from \\eqref{eq:parth} we obtain, for every $\\xi \\geq 0$,\n\\begin{equation} \n\\label{eq:cru00}\na \\, \\xi \\leq \\textsc{f}^{[a,b]}(\\beta, h + \\xi ; 0) -\n\\textsc{f}^{[a,b]}(\\beta, h ; 0) \\leq b \\, \\xi.\n\\end{equation}\nChoosing $\\xi = C^+_{\\beta,\\delta} \\frac{b}{a} \\beta \\delta$ and $\\xi = C^-_{\\beta,\\delta} \n\\frac{a}{b} \\beta \\delta$, respectively, and combining \\eqref{eq:cru0}-\\eqref{eq:cru00}, we \nfinally get the following relation, which holds for all $\\beta, \\delta \\in [0,\\epsilon_0)$, $h\\in\\mathbb{R}$ \nand $0 < a < b < \\infty$:\n\\begin{equation} \n\\label{eq:cru00+}\n\\textsc{f}^{[a,b]} \\big( \\beta, h + C^-_{\\beta,\\delta} \\tfrac{a}{b} \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{[a,b]}(\\beta, h| \\delta) \n\\leq \\textsc{f}^{[a,b]} \\big( \\beta, h + C^+_{\\beta,\\delta} \\tfrac{b}{a} \\beta \\delta ; 0 \\big).\n\\end{equation}\nNext, fix any $x > 0$ and $\\eta > 0$. If $a_n \\uparrow x$ and $b_n \\downarrow x$, then\n$a_n\/b_n \\geq 1-\\eta$ and $b_n\/a_n \\leq 1+\\eta$ for large $n$. Since $h \\mapsto \\textsc{f}^{[a,b]}\n(\\beta, h;\\delta)$ is non-decreasing, by \\eqref{eq:parth} and \\eqref{eq:s0}, \nfor $n$ large enough we have\n\\begin{equation}\n\\textsc{f}^{[a_n,b_n]} \\big( \\beta, h + C^-_{\\beta,\\delta} (1-\\eta) \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{[a_n,b_n]}(\\beta, h; \\delta) \n\\leq \\textsc{f}^{[a_n,b_n]} \\big( \\beta, h + C^+_{\\beta,\\delta} (1+\\eta) \\beta \\delta ; 0 \\big).\n\\end{equation}\nRecalling \\eqref{eq:fx} and \\eqref{eq:fbetah}, we can let $n\\to\\infty$ to get that, for every \n$x > 0$,\n\\begin{equation}\n\\textsc{f}^{\\{x\\}} \\big( \\beta, h + C^-_{\\beta,\\delta} (1-\\eta) \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{\\{x\\}}(\\beta, h; \\delta) \n\\leq \\textsc{f}^{\\{x\\}} \\big( \\beta, h + C^+_{\\beta,\\delta} (1+\\eta) \\beta \\delta ; 0 \\big).\n\\end{equation}\nThis relation also holds for $x=0$ because $\\textsc{f}^{\\{0\\}}(\\beta, h; \\delta)$ is a constant, as we \nshowed in \\eqref{eq:f0}. Taking the supremum over $x \\in [0, s_0]$, we have\nshown that, for all $\\beta, \\delta \\in [0,\\epsilon_0)$ and $h\\in\\mathbb{R}$,\n\\begin{equation}\n\\textsc{f} \\big( \\beta, h + C^-_{\\beta,\\delta} (1-\\eta) \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}(\\beta, h; \\delta) \n\\leq \\textsc{f} \\big( \\beta, h + C^+_{\\beta,\\delta} (1+\\eta) \\beta \\delta ; 0 \\big).\n\\end{equation}\nSince $h \\mapsto \\textsc{f}^{[a,b]}(\\beta, h;\\delta)$ is convex and finite, and hence continuous, we \ncan let $\\eta \\downarrow 0$ to obtain \\eqref{eq:compdeltah} for $\\delta \\in [0,\\epsilon_0)$.\n\n\\medskip\\noindent\n\\textbf{8.}\nThe case $\\delta \\in (-\\epsilon_0, 0]$ is analogous. The inequality in \\eqref{eq:cru0} is replaced by\n\\begin{gather} \n\\label{eq:cru0neg}\nC^-_{\\beta,\\delta} \\, \\beta \\, a \\, (-\\delta) \n\\leq \\textsc{f}^{[a,b]}(\\beta, h; 0) \\,-\\, \\textsc{f}^{[a,b]}(\\beta, h; \\delta)\n\\leq C^+_{\\beta,\\delta} \\, \\beta \\, b \\, (-\\delta),\n\\end{gather}\nwhile \\eqref{eq:cru00} for $\\xi \\leq 0$ becomes\n\\begin{equation} \n\\label{eq:cru00neg}\na \\, (-\\xi)  \\leq\n\\textsc{f}^{[a,b]}(\\beta, h ; 0) - \\textsc{f}^{[a,b]}(\\beta, h + \\xi ; 0) \\leq b \\, (-\\xi).\t\n\\end{equation}\nChoosing $\\xi = C^+_{\\beta,\\delta} \\frac{b}{a} \\beta \\delta$ and $\\xi = C^-_{\\beta,\\delta} \n\\frac{a}{b} \\beta \\delta$, respectively, we get\n\\begin{equation} \n\\label{eq:cru00+neg}\n\\textsc{f}^{[a,b]} \\big( \\beta, h + C^+_{\\beta,\\delta} \\tfrac{b}{a} \\beta \\delta ; 0 \\big) \n\\leq \\textsc{f}^{[a,b]}(\\beta, h; \\delta) \n\\leq \\textsc{f}^{[a,b]} \\big( \\beta, h + C^-_{\\beta,\\delta} \\tfrac{a}{b} \\beta \\delta ; 0 \\big).\n\\end{equation}\nIt remains to let $a \\uparrow x$, $b \\downarrow x$, followed by taking the supremum\nover $x \\in [0,s_0]$\n\n\\medskip\\noindent\n\\textbf{9.}\nFinally, by \\eqref{eq:C+-}, we have $0 < C^-_{\\beta,\\delta} \\le C^+_{\\beta,\\delta} \n< \\infty$ for all $\\beta \\in [0,\\epsilon_0)$ and $\\delta \\in (-\\epsilon_0, \\epsilon_0)$. By \ndominated convergence, $(\\beta,\\delta) \\mapsto c^\\pm_{\\beta,\\delta}$ are continuous \non $[0,\\epsilon_0) \\times (-\\epsilon_0,\\epsilon_0)$, and hence also $(\\beta,\\delta) \\mapsto \nC^\\pm_{\\beta,\\delta}$ is continuous. \nSince $C^\\pm_{0,0} = \\bbvar(\\omega_1) = 1$,\nthe proof is complete.\n\\qed\n\n\\smallskip\n\n\\section{Smoothing with respect to a shift: proof of Theorem~\\ref{th:smoshift}}\n\\label{sec:smoothshift}\n\nEquations \\eqref{eq:ZNgen} and \\eqref{eq:s0} imply that $h \\mapsto \\textsc{f}(\\beta,h;\\delta)$ \nis non-decreasing. Since $\\textsc{f}(\\beta,h;\\delta) \\geq 0$ under Assumption~\\ref{ass:model},\nby \\eqref{eq:polylb}, if $\\textsc{f}(\\bar\\beta, \\bar h; 0) = 0$, then $\\textsc{f}(\\bar\\beta, \\bar h + t; 0) = 0$ \nfor all $t \\leq 0$, and \\eqref{eq:compdeltah2} is trivially satisfied. Henceforth we assume\n$t > 0$.\n\nRecalling the statement of Theorem~\\ref{th:compdeltah}, we set $F_\\beta(\\delta) \n:= C^-_{\\beta,\\delta} \\, \\delta$. This is a continuous and strictly increasing function \nof $\\delta$, with $F_\\beta(0) = 0$, and hence it maps the open interval $(0,\\epsilon_0)$\ninto $(0,\\epsilon'_0)$, for some $\\epsilon'_0 > 0$. Applying the first inequality in \n\\eqref{eq:compdeltah} for $t \\in (0, \\bar\\beta\\epsilon'_0)$, we can write\n\\begin{equation}\n\\textsc{f}(\\bar\\beta, \\bar h + t; 0) \n= \\textsc{f}\\big(\\bar\\beta, \\bar h + \\bar\\beta F_{\\bar\\beta}\n(F_{\\bar\\beta}^{-1}(\\tfrac{t}{\\bar\\beta})); 0\\big) \n\\leq \\textsc{f}\\big(\\bar\\beta,\\bar h; F_{\\bar \\beta}^{-1}(\\tfrac{t}{\\bar\\beta}) \\big).\n\\end{equation}\nApplying \\eqref{eq:smodelta}, we obtain\n\\begin{equation}\n\\textsc{f}(\\bar\\beta, \\bar h + t; 0) \n\\leq \\frac{\\gamma}{2 \\bar\\beta^2} \\, A_{\\bar\\beta,\\frac{t}{\\bar\\beta}} \\, t^2,\n\\end{equation}\nwhere\n\\begin{equation}\nA_{\\beta,\\delta} := B_{F_{\\beta}^{-1}(\\delta)} \\, \n\\bigg(\\frac{F_{\\beta}^{-1}(\\delta)}{\\delta} \\bigg)^2.\n\\end{equation}\nIt follows from \\eqref{eq:propCpm} that\n$\\lim_{(\\beta,\\delta) \\to (0,0)} (F_{\\beta}^{-1}(\\delta)\/\\delta) = 1$.\nSince $\\lim_{\\delta\\to 0} B_\\delta = 1$, we obtain $\\lim_{(\\beta,\\delta) \\to (0,0)} A_{\\beta,\\delta} \n= 1$.\n\\qed\n\n\n\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s1}\n\nWe consider the problem of characterizing positive definite kernels on a product of spheres.\\ The focus will be on continuous and isotropic kernels, keeping the setting originally adopted by I. J. Schoenberg in his influential paper published in 1942 (\\cite{schoen}).\n\nAs usual, let $S^{m}$ denote the unit sphere in the $(m+1)$-dimensional space $\\mathbb{R}^{m+1}$ and $S^\\infty$ the unit sphere in $\\mathbb{R}^\\infty$, the usual real $\\ell^2$ space.\\ Throughout the paper, we will be dealing with real, continuous and isotropic kernels on the product $S^{m}\\times S^{M}$, $m,M=1,2,\\ldots, \\infty$.\\ When speaking of continuity, we will assume each sphere is endowed with its usual geodesic distance.\\ The {\\em isotropy} (zonality) of a kernel $K$ on $S^{m} \\times S^{M}$ refers to the fact that\n$$\nK((x,z),(y,w))=f(x\\cdot y,z \\cdot w), \\quad x,y \\in S^{m},\\quad z,w \\in S^{M},\n$$\nfor some real function $f$ on $[-1,1]^2$, where $\\cdot$ stands for the inner product of both $\\mathbb{R}^{m+1}$ and $\\mathbb{R}^{M+1}$.\\ In particular, the concept introduced above demands the usual notion of isotropy on each sphere involved.\\ In many places in the paper, we will refer to $f$ as the {\\em isotropic part} of $K$.\n\nRecall that if $X$ is a nonempty set, a kernel $K$ is {\\em positive definite} on $X$ if\n$$\n\\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu K(x_\\mu, x_\\nu) \\geq 0,\n$$\nfor $n\\geq 1$, distinct points on $X$, and reals scalars $c_1, c_2, \\ldots, c_n$.\\ In other words, for any $n\\geq 1$ and any distinct points $x_1, x_2, \\ldots, x_n$ on $X$, the $n\\times n$ matrix with entries $K(x_\\mu,x_\\nu)$ is nonnegative definite.\\ In this paper, we will present a characterization for the positive definiteness of a continuous and isotropic kernel on $X=S^m \\times S^M$ based upon Fourier expansions.\n\nIsotropy and positive definiteness for kernels on a single sphere were first considered by I. J. Schoenberg in \\cite{schoen}.\\ He showed that a continuous and isotropic kernel $K$ on $S^{m}$ is positive definite if and only if $K(x,y)=g(x\\cdot y)$, $x,y \\in S^m$, in which the isotropic part $g$ of $K$ has a series representation in the form\n$$\ng(t)=\\sum_{k=0}^{\\infty} a_k^m P_k^{m}(t),\\quad t \\in [-1,1],\n$$\nin which $a_k^m \\geq 0$, $k \\in \\mathbb{Z}_+$ and $\\sum_{k=0}^{\\infty}a_k P_k^{m}(1) <\\infty$.\\ The symbol $P_k^{m}$ stands for the usual\nGegenbauer polynomial of degree $k$ associated with the rational $(m-1)\/2$, as discussed in \\cite{szego}.\\ This Schoenberg's outstanding result is far-reaching and has ramifications in distance geometry, statistics, spherical designs, approximation theory, etc.\\ In approximation theory, positive definite kernels are used in interpolation of scattered data over the sphere.\\ The importance of this problem in many areas of science and engineering is reflected in the literature, where different methods to solve such problem have been proposed.\\ Given $n$ distinct data points $x_1, x_2, \\ldots, x_n$ on $S^{m}$ and a target\nfunction $h:S^{m} \\to \\mathbb{R}$,\nthe interpolation problem itself requires the finding of a continuous function $s:S^{m} \\to \\mathbb{R}$ of the form\n$$\ns(x)=\\sum_{j=1}^n \\lambda_j g(x \\cdot x_j),\\quad x \\in S^{m},\\quad \\lambda_1, \\lambda_2, \\ldots, \\lambda_n \\in \\mathbb{R},$$\nso that $s(x_i)=h(x_i)$, $i=1,2,\\ldots,n$.\nIf we choose the prescribed function $g$ to be the isotropic part of a convenient positive definite kernel $K$, then the interpolation problem has a unique solution for any $n$ and any $n$ data points.\n\nAn outline of the paper is as follows.\\ In Section 2, we present several technical results that culminate with a characterization for the continuous, isotropic and positive definite kernels on $S^m \\times S^M$, $m,M <\\infty$.\\ In Section 3, we complete this circle of ideas, by reaching a similar characterization in the cases in which at least one of the spheres involved is the real Hilbert sphere $S^\\infty$.\\ Finally, Section 4 contains a few relevant remarks along with the description of future lines of investigation on the subject.\n\n\\section{Positive definiteness on $S^{m}\\times S^{M}$, $m,M < \\infty$.}\\label{s2}\n\nThis section contains all the technical material needed in the proof of the extension of Schoenberg's theorem to a product of spheres $S^m \\times S^M$, in the case when both $m$ and $M$ are finite.\\ The proof will require a series of well-known results involving Gegenbauer polynomials and also a few facts from the analysis on the sphere.\\ We suggest the classical reference \\cite{szego} for the first topic and \\cite{atkinson,dai,groemer,muller} for the other.\n\nThe orthogonality relation for Gegenbauer polynomials reads as follows (\\cite[p.10]{dai}):\n$$\n\\int_{-1}^1 P_n^{m}(t) P_k^{m}(t) (1-t^2)^{(m-2)\/2}dt=\\frac{\\tau_{m+1}}{\\tau_{m}}\\frac{m-1}{2n+m-1}P_n^{m}(1)\\delta_{n,k},\n$$\nin which $\\tau_{m+1}$ is the surface area of $S^{m}$, that is,\n$$\n\\tau_{m+1}:=\\frac{2\\pi^{(m+1)\/2}}{\\Gamma((m+1)\/2)}.\n$$\n\nSince Schoenberg's characterization for positive definiteness on $S^{m}$ is based upon Fourier expansions with respect to the orthogonal family $\\{P_n^{m}:n=0,1,\\ldots\\}$, it is quite natural to expect that a similar characterization for positive definiteness on $S^{m}\\times S^{M}$ will require expansions with respect to the tensor family\n$$\\{(t,s) \\in [-1,1]^2 \\to P_k^m(t)P_l^M(s): k,l=0,1,\\ldots\\}.$$\n\nThe first important fact to be noticed about the functions in the family above is this.\n\n\\begin{lem} \\label{gegpd}\nIf $k,l \\in \\mathbb{Z}_+$, then $(t,s) \\in [-1,1]^2 \\to P_k^m(t)P_l^M(s)$ is the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$.\n\\end{lem}\n\\pf This follows from the definition of positive definiteness, Schoenberg's original characterization for positive definite kernels and the Schur product theorem (\\cite[p.458]{horn}).\\ The later asserts that the entry-wise product of two nonnegative definite matrices of same order is a nonnegative definite matrix itself.\\eop\n\nThe tensor family is orthogonal on $[-1,1]^2$ with respect to the weight function\n$$\nw_{m,M}(t,s)=(1-t^2)^{(m-2)\/2}(1-s^2)^{(M-2)\/2},\\quad t,s \\in [-1,1].\n$$\nThe $(k,l)$-Fourier coefficient of a function $f: [-1,1]^2 \\to \\mathbb{R}$ from $L^1([-1,1]^2, w_{m,M})$\nis\n$$\n\\hat{f}_{k,l}:=\\frac{1}{\\tau_k^m \\tau_l^M}\\int_{[-1,1]^2} f(t,s)P_k^{m}(t)P_l^{M}(s)dw_{m,M}(t,s), \\quad k,l \\in \\mathbb{Z}_+,\n$$in which\n$$\n\\tau_k^m:=\\frac{\\tau_{m+1}}{\\tau_{m}}\\frac{m-1}{2k+m-1}P_k^{m}(1),\\quad k\\in \\mathbb{Z}_+.\n$$\nThe next lemma describes an alternative way for computing these Fourier coefficients.\\ The symbol $\\sigma_m$ will denote the surface measure on $S^m$.\n\n\\begin{lem} \\label{pdmultiple} If $f$ belongs to $L^1([-1,1]^2,w_{m,M})$, then the Fourier coefficient $\\hat{f}_{k,l}$ is a positive constant multiple of\n$$\n\\int_{S^{m}\\times S^{M}} \\left[\\int_{S^{m}\\times S^{M}} f(x\\cdot y,z \\cdot w)P_k^{m}(x\\cdot y) \\times P_l^{M}(z\\cdot w)d\\sigma_{m}(y)d\\sigma_{M}(w)\\right]d\\sigma_{m}(x)d\\sigma_{M}(z).\n$$\n\\end{lem}\n\\pf If $m,M \\geq 2$, it suffices to employ the Funk-Hecke formula (\\cite[p.11]{dai}) in the expression defining the Fourier coefficient.\\\nThe Funk-Hecke formula states that\n$$\n\\int_{S^{M}}g(z\\cdot w)P_l^{M}(z\\cdot w)d\\sigma_{M}(w)=\\tau_{m-1}\\int_{-1}^1 g(s)P_l^{M}(s)(1-s^2)^{(M-2)\/2}ds,\\quad z \\in S^{M},\n$$\nwhenever $l\\in \\mathbb{Z}_+$ and $g \\in L^2([-1,1],w_M)$.\\ Using the formula with\n$$\ng(s)=\\int_{-1}^1 f(t,s)P_k^{m}(t)(1-t^2)^{(m-2)\/2}dt, \\quad s \\in [-1,1],\n$$\nit is promptly seen that $\\hat{f}_{k,l}$ is a positive multiple of\n$$\n\\int_{S^{M}} \\int_{-1}^1 f(t,z \\cdot w)P_k^{m}(t)(1-t^2)^{(m-2)\/2}dt \\ P_l^{M}(z\\cdot w) d\\sigma_{M}(w).\n$$\nApplying a similar argument in the internal integral reveals that $\\hat{f}_{k,l}$ is a positive multiple of\n$$\n\\int_{S^{M}} \\int_{S^{m}} f(x\\cdot y,z \\cdot w)P_k^{m}(x\\cdot y) P_l^{M}(z\\cdot w)d\\sigma_{m}(y)d\\sigma_{M}(w).\n$$\nIntegration with respect to the remaining variables concludes the proof.\\ In the cases in which either $m=1$ or $M=1$, the arguments demand the replacement of the Funk-Hecke formula with direct computation.\\eop\n\n\\begin{lem} \\label{pdtimes} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then\n$$\n\\int_{S^{m}\\times S^{M}}\\left[\\int_{S^{m}\\times S^{M}}f(x\\cdot y,z\\cdot w)d\\sigma_{m}(x)d\\sigma_{M}(z)\\right]d\\sigma_{m}(y)d\\sigma_{M}(w) \\geq 0.\n$$\n\\end{lem}\n\\pf It suffices to write the double integral $I$ in the statement of the theorem as a double limit of Riemann sums.\\ Indeed, we can select a sequence $\\{\\mathcal{P}_n: n=0,1,\\ldots\\}$ of partitions of $S^{m}\\times S^{M}$\nin such a way that $\\mathcal{P}_n=\\{Q_1^n,Q_2^n,\\ldots, Q_{\\alpha(n)}^n\\}$, the sequence $\\{\\alpha(n)\\}$ increases to $\\infty$ and the sequences of diameters $\\{\\mbox{diam}(Q_j^n)\\}$ satisfy\n$\\lim_{n\\to \\infty}\\mbox{diam}(Q_j^n)=0$.\\ Picking points $(x_j^n,z_j^n) \\in Q_j^n$, we can write\n$$\nI=\\lim_{N \\to \\infty} \\sum_{J=1}^{\\alpha(N)}\\left[ \\int_{S^{m}\\times S^{M}}f(x \\cdot x_J^N,z \\cdot z_J^N)d\\sigma_{m}(x)d\\sigma_{M}(z)\\right]\\mbox{vol}(Q_J^N).\n$$\nRepeating the procedure with the resulting integral leads to\n\\begin{eqnarray*}\nI & = & \\lim_{N \\to \\infty} \\sum_{J=1}^{\\alpha(N)}\\left[\\lim_{n \\to \\infty}\\sum_{j=1}^{\\alpha(n)}f(x_j^n \\cdot x_J^N,z_j^n \\cdot z_J^N) \\mbox{vol}(Q_j^n)\\right] \\mbox{vol}(Q_J^N)\\\\\n  & = & \\lim_{N \\to \\infty} \\lim_{n \\to \\infty}\\sum_{J=1}^{\\alpha(N)}\\sum_{j=1}^{\\alpha(n)}\\mbox{vol}(Q_j^n)\\mbox{vol}(Q_J^N)f(x_j^n \\cdot x_J^N,z_j^n \\cdot z_J^N).\n  \\end{eqnarray*}\nSince the double limit above exists, it follows that\n$$\nI=\\lim_{n \\to \\infty}\\sum_{j,J=1}^{\\alpha(n)}\\mbox{vol}(Q_j^n)\\mbox{vol}(Q_J^n)f(x_j^n \\cdot x_J^n,z_j^n \\cdot z_J^n).\n$$\nIf $f$ is the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then each double sum in the last expression above is clearly nonnegative.\\ In particular, the limit itself is nonnegative as well.\\eop\n\nWe now combine the three lemmas above in order to obtain the following result.\n\n\\begin{lem} \\label{coeffpos} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$ then\n$$\n\\hat{f}_{k,l} \\geq 0, \\quad k,l \\in \\mathbb{Z}_+.\n$$\n\\end{lem}\n\\pf Let us fix $k$ and $l$.\\ Lemma \\ref{gegpd} and the Schur product theorem guarantees that the function\n$$\n(t,s) \\in [-1,1]^2 \\to f(t,s)P_k^{m}(t)P_l^{M}(s)\n$$\nis the continuous and isotropic part of a positive definite kernel on $S^{m} \\times S^{M}$.\\ Taking into account this information and that one provided by Lemma \\ref{pdmultiple}, an application of Lemma \\ref{pdtimes} leads to the inequality in the statement of the lemma.\\eop\n\nNext, we recall one of the several generating formulas for the Gegenbauer polynomials, the Poisson identity (\\cite[p.419]{dai}).\n\n\\begin{lem} \\label{poisson} If $r \\in [0,1)$, then\n$$\n\\frac{1-r^2}{(1-2tr+r^2)^{(m+1)\/2}}=\\sum_{k=0}^{\\infty}\\frac{2k+m-1}{m-1}P_k^{m}(t)r^k, \\quad t\\in [-1,1].\n$$\nIf $r_0 \\in [0,1)$ is fixed, then the convergence of the series is absolute and uniform for $(r,t) \\in [0,r_0]\\times [-1,1]$.\n\\end{lem}\n\nWe are about ready to prove the following auxiliary result.\n\n\\begin{lem} \\label{spoisson} Let $f$ be the continuous and isotropic part of a kernel on $S^{m}\\times S^{M}$.\\ If $r,\\rho \\in [0,1)$, then the double series\n$$\n\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)r^k\\rho^l\n$$\nconverges.\\ As a matter of fact, there exists a positive constant $C$, depending upon $f$ only, so that\n$$\n\\left|\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)r^k\\rho^l\\right|\\leq C, \\quad r,\\rho \\in [0,1).\n$$\n\\end{lem}\n\\pf Let $a_{k,l}^{r,\\rho}$ denote the general term of the series in the statement of the lemma.\\ It is promptly seen that\n$$\na_{k,l}^{r,\\rho}=C_1\\int_{-1}^1 \\int_{-1}^1 f(t,s)\\frac{2k+m-1}{m-1}P_k^{m}(t)r^k\\frac{2l+M-1}{M-1}P_l^{M}(t)\\rho^l dw_{m,M}(t,s),\n$$\nin which\n$$\nC_1=\\frac{\\tau_{m}\\tau_{M}}{\\tau_{m+1}\\tau_{M+1}}.\n$$\nOn the other hand, Lemma \\ref{poisson} implies that\n$$\n\\sum_{k=0}^{\\infty}\\sum_{l=0}^{\\infty}a_{k,l}^{r,\\rho}=C_1 \\int_{-1}^1 \\int_{-1}^1f(t,s)\\frac{1-r^2}{(1-2rt+r^2)^{(m+1)\/2}}\\frac{1-\\rho^2}{(1-2\\rho s+s^2)^{(M+1)\/2}}dw_{m,M}(t,s),\n$$\nwhenever $r,\\rho \\in [0,1)$.\\ Thus, since $f$ is continuous and the left hand side of the Poisson identity is positive, the proof of the first half of the lemma reduces itself to proving that the double integral\n$$\n\\int_{-1}^1\\int_{-1}^1 \\frac{1-r^2}{(1-2rt+r^2)^{(m+1)\/2}}\\frac{1-\\rho^2}{(1-2\\rho s+s^2)^{(M+1)\/2}}(1-t^2)^{(m-2)\/2}(1-s^2)^{(M-2)\/2}dtds\n$$\nis finite.\\ But, this follows from the well-known property of the Poisson kernels (\\cite[p.47]{muller})\n$$\n\\int_{-1}^1 \\frac{1-r^2}{(1-2rt+r^2)^{(m+1)\/2}}(1-t^2)^{(m-2)\/2}dt=\\frac{\\tau_{m+1}}{\\tau_{m}}.\n$$\nIn particular, $C:=\\max\\{|f(s,t)|: -1\\leq t,s\\leq 1\\}$ fits into what is needed in the second statement of the lemma.\\eop\n\n\\begin{lem}\\label{pd1} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then the double series\n$$\n\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)\n$$\nconverges.\n\\end{lem}\n\\pf Let $f$ be the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$.\\ Due to Lemma \\ref{coeffpos}, we know already that all the Fourier coefficients $\\hat{f}_{k,l}$ are nonnegative.\\ In particular, the double sequence $\\{s_{p,q}\\}$ of partial sums of the double series in the statement of the current lemma is monotonically increasing, that is, $s_{p,q} \\leq s_{\\mu,\\nu}$ when $p\\leq \\mu$ and $q \\leq \\nu$.\\ On the other, the previous lemma produces the inequality\n$$\n\\sum_{k=0}^{p}\\sum_{l=0}^{q}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)r^k\\rho^l \\leq C, \\quad p,q\\in\\mathbb{Z}_+, \\quad r,\\rho \\in [0,1),\n$$\nfor some $C>0$.\\ By taking a double limit when $r,\\rho \\to 1^{+}$, we deduce that the double sequence of partial sums $\\{s_{p,q}\\}$ is bounded above.\\ A classical result from the theory\nof double sequences (\\cite[p.373]{limaye}) implies that $\\{s_{p,q}\\}$ converges, that is, the series in the statement of the lemma converges.\\eop\n\nThe Weierstrass M-test can be adapted to hold for double series of functions.\\ Combining it with Lemma \\ref{pd1} leads to the proposition below.\n\n\\begin{prop}\\label{conve} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, then\n$$\n\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s)\n$$\nconverges absolutely and uniformly for $(t,s) \\in [-1,1]^2$.\n\\end{prop}\n\nThe main result in this section is as follows.\n\n\\begin{thm} \\label{mainPD} Let $K$ be a continuous and isotropic kernel on $S^{m}\\times S^{M}$.\\ It is positive definite on $S^{m}\\times S^{M}$ if and only if its isotropic part $f$ has a representation in the form\n$$\nf(t,s)=\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nin which $\\hat{f}_{k,l} \\geq 0$, $k,l \\in \\mathbb{Z}_+$ and $\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(1)P_{l}^{M}(1)<\\infty$.\n\\end{thm}\n\\pf If the isotropic part $f$ of $K$ has the representation announced in the theorem, the series appearing there is uniformly and absolutely convergent.\\ In particular, Lemma \\ref{gegpd} implies that $f$ is\na pointwise double limit of functions which are isotropic parts of positive definite kernels on $S^{m} \\times S^{M}$.\\ Consequently, $K$ itself is positive definite on $S^{m}\\times S^{M}$.\\ Conversely, assume $K$ is positive definite on $S^{m}\\times S^{M}$ and write $f$ to denote its isotropic part.\\ Lemma \\ref{pd1} and Proposition \\ref{conve} supply a function $g$ so that\n$$\ng(s,t)=\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}P_{k}^{m}(t)P_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nwith uniform convergence in $[-1,1]^2$.\\ In particular, $g$ is continuous in $[-1,1]^2$.\\ On the other hand, the same uniform convergence and the\northogonality relation mentioned at the beginning of the section imply that\n$$\n\\hat{f}_{k,l} -\\hat{g}_{k,l}=0, \\quad k,l \\in \\mathbb{Z}_+.\n$$\nConsequently, $f=g$.\\eop\n\n\\section{Positive definiteness on $S^\\infty \\times S^M$}\n\nIn this section, we extend Theorem \\ref{mainPD} to the cases in which either $m=\\infty$ or $M=\\infty$.\\ Clearly, it suffices to consider the cases $m=\\infty$, $M<\\infty$ and $m=M=\\infty$ only.\n\nEvery sphere $S^{m}$ can be isometrically embedded in $S^{\\infty}$.\\ In particular, a positive definite kernel on $S^\\infty \\times S^{M}$ is positive definite on $S^{m}\\times S^{M}$, for $m=1,2,\\ldots$.\\ Likewise, if $f$ is the isotropic part of a positive definite kernel on $S^\\infty \\times S^{M}$, then it is the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$, for $m=1,2, \\ldots$.\\ In addition, if $f$ is continuous, then for\nevery $m\\geq 1$, we have a representation for $f$ in the form\n$$\nf(t,s)=\\sum_{k,l=0}^{\\infty}\\hat{f}_{k,l}^{m,M} P_{k}^{m}(t) P_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nin which\n$$\n\\hat{f}_{k,l}^{m,M} =\\frac{1}{\\tau_{k}^{m} \\tau_{l}^{M}}  \\int_{[-1,1]^{2}} f(x,y) P_{k}^{m}(t)P_{l}^{M}(s) dw_{m,M}(t,s)\\geq 0,\\quad k,l \\in \\mathbb{Z}_+.\n$$\nand $\\sum_{k=0}^{\\infty}\\sum_{l=0}^\\infty \\hat{f}_{k,l}^{m,M}P_{k}^{m-1}(1) P_{l}^{M-1}(1)<\\infty$.\\ Below, we will prefer to normalized the above expressions by writing\n$$\nR_k^{m}=\\frac{P_k^{m}}{P_k^{m}(1)}, \\quad k\\in \\mathbb{Z}_+,\n$$\nand\n$$\nf(t,s)=\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{m,M} R_{k}^{m}(t) R_{l}^{M}(s), \\quad t,s \\in [-1,1],\n$$\nwhere now\n$$\n\\check{f}_{k,l}^{m,M}=P_k^{m}(1)P_l^{M}(1)\\hat{f}_{k,l}^{m,M}, \\quad k,l \\in \\mathbb{Z}_+.\n$$\nBefore we proceed, it is convenient to mention that the Fourier coefficients introduced above are well-defined as long as $f$ belongs to $L^1([-1,1],w_{m,M})$.\n\n\\begin{lem}\\label{pre} Let $f$ belong to $L^1([-1,1],w_{m,M})$.\\ If $k$ and $l$ are fixed nonnegative integers, then the sequence $\\{\\check{f}_{k,l}^{2m,M}:m=1,2,\\ldots\\}$ is convergent.\n\\end{lem}\n\\pf Using the following recurrence relation for Gegenbauer polynomials (\\cite[p. {84}]{szego})\n$$ (1-t^{2})P^{m+2}_{k}(t) = \\frac{(k+m-1)(k+m)}{(m-1)(2k+m+1)} P_{k}^{m}(t) - \\frac{(k+1)(k+2)}{(m-1)(2k+m+1)}P_{k+2}^{m}(t),\n$$\nit is easy to deduce that\n$$\n\\check{f}_{k,l}^{m+2,M} = \\frac{(k+m-1)(k+m)}{m(2k+m-1)} \\check{f}_{k,l}^{m,M} - \\frac{(k+1)(k+2)}{m(2k+m+3)} \\check{f}_{k+2,l}^{m,M},\\quad m\\geq 1.\n$$\nConsequently,\n\\begin{align*}\n|\\check{f}_{k,l}^{m+2,M} -  \\check{f}_{k,l}^{m,M} | & = \\left |\\frac{k(k-1)}{m(2k+m-1)}\\check{f}_{k,l}^{m,M} - \\frac{(k+1)(k+2)}{m(2k+m+3)} \\check{f}_{k+2,l}^{m,M} \\right | \\\\\n& \\leq \\left[ \\frac{k(k-1)}{m(2k+m-1)} + \\frac{(k+1)(k+2)}{m(2k+m+3)}\\right]f(1,1),\\quad m\\geq 1.\n\\end{align*}\nAs an obvious consequence, $\\{\\check{f}_{k,l}^{2m,M}\\}$ is a Cauchy sequence of real numbers, therefore, convergent. \\eop\n\n\\begin{lem}\\label{prepar}\nIf $f$ is the continuous and isotropic part of a positive definite kernel on $S^{\\infty} \\times S^{M}$, then the double series\n$$\n\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{m,M}\\, t^k R_{l}^{M}(s)\n$$\nconverges for $(t,s) \\in (-1,1)^2$, uniformly in $m$.\n\\end{lem}\n\\pf In order to see that the series converges in $(-1,1)^2$, for every $m$, it suffices to show that $\n\\sum_{k=0}^{\\infty}\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M} |t|^k$ converges.\\ Recalling Tonelli's theorem for convergence of double series (\\cite[p.384]{limaye}),\nthat will follow as long as\n$\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M}$ converges for all $k$ and the iterated series $\n\\sum_{k=0}^{\\infty}\\left(\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M}\\right) |t|^k $ converges.\\ But both assertions follow from the inequalities\n$$\n\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M} \\leq \\sum_{\\mu,l=0}^\\infty  \\check{f}_{\\mu,l}^{m,M}= f(1,1), \\quad k=0,1,\\ldots,\n$$\nand\n$$\n\\sum_{k=0}^{\\infty}\\left(\\sum_{l=0}^\\infty \\check{f}_{k,l}^{m,M}\\right) |t|^k \\leq f(1,1)\\sum_{k=0}^{\\infty}|t|^k =\\frac{f(1,1)}{1-|t|}, \\quad t\\in (-1,1).\n$$\nAs for the uniform convergence in $m$, it suffices to observe that\n$$\n\\sum_{k,l=0}^{\\infty}\\check{f}_{k,l}^{m,M}\\, t^k R_{l}^{M}(s) \\leq f(1,1)\\sum_{k=0}^{\\infty}|t|^k \\quad t,s \\in (-1,1),\n$$\nThe proof is complete.\\eop\n\nThe next lemma is a technical result that can be found proved in \\cite{schoen}.\n\n\\begin{lem}\\label{limitscho}\nIf $t \\in (-1,1)$, then the sequence $\\{R_k^m(t)\\}$ converges to $t^k$ as $m \\to \\infty$, uniformly in $k$.\n\\end{lem}\n\n\\begin{thm} \\label{maininfty} Let $K$ be a continuous and isotropic kernel on $S^{\\infty}\\times S^{M}$.\\ It is positive definite on $S^{\\infty}\\times S^{M}$ if and only if its isotropic part $f$ has a representation in the form\n$$ f(t,s) = \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{M}t^{k}R_l^{M}(s),$$\nin which $\\check{f}_{k,l}^{M} \\geq 0$, $k,l \\in \\mathbb{Z}_+$ and $\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{M} < \\infty$.\n\\end{thm}\n\\pf For each $k$ and $l$, the function $(t,s)\\in [-1,1]^2 \\to t^{k}R_l^{M}(s)$ is the isotropic part of a positive definite kernel on $S^\\infty \\times S^{M}$.\\ Hence, if $f$ has the representation described in the statement of the theorem, then $K$ is a pointwise limit of positive definite kernels.\\ In particular, it is positive definite itself.\\ Conversely, assume $K$ is positive definite.\\ Without loss of generality, we can assume that $K$ is nonzero.\\ Hence, we can assume that its isotropic part $f$ satisfies $f(1,1) >0$.\\ Since $f$ is the isotropic part of a positive definite kernel on each product $S^{m} \\times S^{M}$, then for each pair $(k,l)$, we may consider the sequence of normalized Fourier coefficients $\\{\\check{f}_{k,l}^{m,M}\\}$.\\ Lemma \\ref{pre} authenticates the definition\n$$\n\\check{f}_{k,l}^{\\, M}:= \\lim_{m \\to \\infty}\\check{f}_{k,l}^{2m,M}, \\quad k,l=0,1,\\ldots.\n$$\nwhile Lemma \\ref{prepar} guarantees that\n$$\n\\lim_{m\\to \\infty}\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M}\\, t^k R_{l}^{M}(s)=\\sum_{k,l=0}^{\\infty}\\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s), \\quad t,s \\in (-1,1).\n$$\nTo proceed, we fix $(t,s) \\in (-1,1)^2$ and $\\epsilon >0$.\\ From the previous limit, we can select $m_0$ so that\n$$\n\\left|\\sum_{k,l=0}^{\\infty}\\check{f}_{k,l}^{2m,M}\\, t^k R_{l}^{M}(s)- \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s)\\right| < \\frac{\\epsilon}{2}, \\quad m\\geq m_0.$$\nBy Lemma \\ref{limitscho}, we can select $m_1$ so that\n$$|R_k^{2m}(t)-t^k| <\\frac{\\epsilon}{2f(1,1)}, \\quad k=0,1,\\ldots, \\quad m\\geq m_1.$$\nIt is now clear that\n\\begin{eqnarray*}\n\\left|\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M}\\, R_k^{2m}(t)R_{l}^{M}(s)-\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M}\\, t^k R_{l}^{M}(s)\\right| & < &  \\frac{\\epsilon}{2f(1,1)}\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{2m,M} \\\\\n& \\leq & \\frac{\\epsilon}{2}, \\quad m\\geq m_1.\n\\end{eqnarray*}\nThus, with the help of an arbitrarily large $m$, we can use both implications above to deduce that\n$$\n0\\leq \\left| f(t,s)- \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s)\\right| < \\epsilon.\n$$\nHence,\n$$\nf(t,s)= \\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}\\, t^k R_{l}^{M}(s), \\quad t,s \\in (-1,1).\n$$\nThe coefficients in the representation above are obviously nonnegative.\\ If $\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{\\, M}$ were not convergent, we could select\na positive integer $N$ so that\n$$\n\\sum_{k,l=0}^{N} \\check{f}_{k,l}^{M} \\geq 2f(1,1).\n$$\nPicking a $\\tau \\in (0,1)$ so that $\\tau^{N} > 1\/2$, we would reach\n$$ f(\\tau,1)=\\sum_{k,l=0}^{\\infty} \\check{f}_{k,l}^{M}\\tau^{k} \\geq  \\sum_{k,l=0}^{N} \\check{f}_{k,l}^{M}\\tau^{k} > f(1,1),$$\na contradiction with the positive definiteness of $f$.\\ Having guaranteed the uniform convergence of the series in the representation for $f$ above and invoking the continuity of $f$ in $[-1,1]^2$, we now can let $t,s\\to 1^{-}$ and $t,s \\to -1^{+}$\nin the representation formula in order to conclude that it also holds in $[-1,1]^2$.\\eop\n\nA standard adaptation of the arguments used in the proof of the previous theorem is all that is needed in order to deduce the following complement.\n\n\\begin{thm}  Let $K$ be a continuous and isotropic kernel on $S^{\\infty}\\times S^{\\infty}$.\\ It is positive definite on $S^{\\infty}\\times S^{\\infty}$ if and only if its isotropic part $f$ has a representation in the form\n$$ f(t,s) = \\sum_{k,l=0}^{\\infty}  f_{k,l}t^{k}s^l,$$\nin which $f_{k,l} \\geq 0$, $k,l \\in \\mathbb{Z}_+$ and $\\sum_{k,l=0}^{\\infty} f_{k,l} < \\infty$.\n\\end{thm}\n\n\n\\section{Final remarks}\n\nIn view of the characterization for the continuous, isotropic and positive definite kernels on a product of the form $S^m \\times S^M$ obtained in the previous section, one may ask what are the other relevant questions regarding that class of kernels.\\ We will mention a few of them in this final section of the paper along with some additional elementary results.\n\nLet us begin with the strictly positive definite kernels.\\ A continuous, isotropic and positive definite kernel $K$ on $S^m \\times S^M$ is {\\em strictly positive definite of order $n$} on $S^m \\times S^M$ if\nits isotropic part $f$ satisfies\n$$\n\\sum_{\\mu=1}^n\\sum_{\\nu=1}^n c_\\mu c_\\nu f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu) >0,\n$$\nwhenever the $n$ points $(x_1, w_1), (x_2, w_2), \\ldots, (x_n,w_n)$ of $S^m \\times S^M$ are distinct and the scalars $c_\\mu$ are not all zero.\\ So, for a fixed $n$, an interesting question would be to characterize, via the main theorems proved here, the continuous, isotropic and strictly positive definite kernels of order $n$ on $S^m \\times S^M$.\\ Going one step further, to characterize the continuous, isotropic and strictly positive definite kernels of all orders on $S^m \\times S^M$.\\ Even in the case of a single sphere, similar characterizations are not available for all fixed $n$ (see \\cite{mene0}).\n\nConcerning the problems mentioned above, the intermediate problem to be described below could provide clues to a complete solution.\\ A continuous, isotropic and positive definite kernel $K$ on $S^m \\times S^M$ is {\\em $DC$-strictly positive definite of order $n$} on $S^m \\times S^M$ if\nits isotropic part $f$ satisfies\n$$\n\\sum_{\\mu=1}^n\\sum_{\\nu=1}^n c_\\mu c_\\nu f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu) >0,\n$$\nwhenever the $n$ points $x_1$, $x_2, \\ldots, x_n$ of $S^m$ are distinct, the $n$ points $w_1, w_2, \\ldots, w_n$ of $S^M$ are distinct and the scalars $c_\\mu$ are not all zero.\\ Obviously, a strictly positive definite kernel of order $n$ on $S^m \\times S^M$ is $DC$-strictly positive definite of order $n$ on $S^m \\times S^M$, but not conversely (unless $n=1$).\\ Thus, to characterize the continuous, isotropic and positive definite kernels on $S^m \\times S^M$ which are $DC$-strictly positive definite of order $n$ on $S^m \\times S^M$ would be an interesting problem as well.\n\nA third problem we would like to mention is the description of consistent methods to construct continuous, isotropic and (strictly) positive definite kernels on $S^m \\times S^M$.\\ In particular, methods based on the description via known classes of continuous, isotropic and (strictly) positive definite kernels on a single sphere.\n\nA very elementary one is this.\n\n\\begin{prop} \\label{prodi} If $f$ is the continuous and isotropic part of a positive definite kernel on $S^{m}$ and $g$ is the continuous and isotropic part of a positive definite kernel on $S^{M}$, then the function $h$ given by the formula\n$$\nh(t,s)=f(t)g(s), \\quad t,s \\in [-1,1],\n$$\nis the isotropic part of a positive definite kernel on $S^{m}\\times S^{M}$.\\ Further, if $f$ is the isotropic part of a strictly positive definite kernel of order $n$ on $S^{m}$ (respect., $g$ is the isotropic part of a strictly positive definite kernel of order $n$ on $S^{M}$) and $g(1)>0$ (respect., $f(1)>0$), then $h$ is the isotropic part of a strictly positive definite kernel of order $n$ on $S^{m} \\times S^{M}$.\n\\end{prop}\n\\pf The first assertion of the theorem is a consequence of the Schur product theorem.\\ As for the second one, it follows from Oppenheim's inequality (\\cite[p.480]{horn}).\\eop\n\nIf the intention is a more concrete example, one may employ completely monotonic functions in two variables.\\ A continuous function $g: [0,\\infty)^2 \\to \\mathbb{R}$ is \\emph{completely monotonic} on $(0,\\infty)^2$ if it is $C^\\infty$ in $(0,\\infty)^2$ and\n$$\n(-1)^{n_1+n_2} \\frac{\\partial^{n_1+n_2} g}{\\partial u^{n_1}\\partial v^{n_2}}(u,v) \\geq 0, \\quad u,v>0, \\quad n_1,n_2\\in\\mathbb{Z}_+.\n$$\nIt is known that function $g$ as above can be represented in the form\n\\begin{eqnarray} \\label{eq-function CM}\ng(u,v) = \\int_{[0,\\infty)^2}e^{-tu-sv}d\\rho(t,s),\\quad u,v >0,\n\\end{eqnarray}\nin which $\\rho$ is a $\\sigma$-additive and nonnegative measure on $[0,\\infty)^2$ satisfying $0<\\rho((0,\\infty)^2)\\leq\\rho([0,\\infty)^2)\\leq\\infty$ (\\cite[p. 87]{bochner}).\n\nA positive scalar multiple of a completely monotonic function on $(0,\\infty)^2$ is itself completely monotonic on $(0,\\infty)^2$.\\ Likewise, the sum and product of two completely monotonic function\non $(0,\\infty)^2$ are completely monotonic on $(0,\\infty)^2$.\\ If $g,h: [0,\\infty) \\to \\mathbb{R}$ are usual completely monotonic functions on $(0,\\infty)$, then $F(u,v)=g(u)h(v)$ is completely monotonic\non $(0,\\infty)^2$.\\ In particular, $(u,v)\\in [0,\\infty)^2\\to \\exp(-u)\\exp(-v)$\n and $(u,v)\\in [0,\\infty)^2\\to 1\/(1+u)^\\alpha(1+v)^\\beta$, $\\alpha,\\beta \\geq 0$, are completely monotonic on $(0,\\infty)^2$.\\ Additional examples can be found in \\cite{samko}.\n\nFor actual examples of positive definite kernels on $S^m \\times S^M$, the following result is quite useful.\n\n\\begin{prop}\nIf $g$ is completely monotonic on $(0,\\infty)^2$, then\n$$\nf(t,s):= g(\\arccos t,\\arccos s)\n$$\nis the isotropic part of a positive definite kernel on $S^m\\times S^M$.\\ Further, if $g$ is nonconstant, then $f$ is the isotropic part of a strictly positive definite kernel on $S^m \\times S^M$.\n\\end{prop}\n\\pf Consider the integral representation for $g$ as described above.\\ If $x_1,\\ldots,x_n\\in S^m$, $w_1,\\ldots,w_n\\in S^M$ and $c_1,\\ldots,c_n$ are real scalars, then\n$$\n\\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu  f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu)  = \\int_{[0,\\infty)^2} \\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu e^{-t\\arccos(x_\\mu\\cdot x_\\nu)-s\\arccos(w_\\mu\\cdot w_\\nu)}d\\rho(t,s),\n$$\nthat is,\n$$\n\\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu  f(x_\\mu \\cdot x_\\nu, w_\\mu \\cdot w_\\nu)  = \\int_{[0,\\infty)^2} \\sum_{\\mu,\\nu=1}^n c_\\mu c_\\nu e^{-t d_m(x_\\mu\\cdot x_\\nu)-sd_M(w_\\mu\\cdot w_\\nu)}d\\rho(t,s),\n$$\nin which $d_m$ and $d_M$ are the usual geodesic distances on $S^m$ and $S^M$, respectively.\\ A result proved in \\cite{alexander} reveals that $d_m$ and $d_M$ are kernels of negative type.\\ Consequently, the matrices with entries $-t d_m(x_\\mu,x_\\nu)-sd_M(w_\\mu,w_\\nu)$ is almost nonnegative definite (\\cite[p.135]{don}).\\ A classical result from the theory of positive definite kernels (\\cite[p.74]{berg}) now implies that $\\exp(-td_m-sd_M)$ is a positive definite kernel on $S^m \\times S^M$.\\ Thus, the initial quadratic form is nonnegative and the first assertion of the proposition is proved.\\ As for the second one, it suffices to observe that if the points $(x_\\mu,w_\\mu)$ are distinct then the matrix with entries $-t d_m(x_\\mu,x_\\nu)-sd_M(w_\\mu,w_\\nu)$ has no pair of identical rows when $t,s>0$.\\ In that case, the kernel $(x,z,y,w) \\in (S^m \\times S^M)^2 \\to \\exp[-td_m(x,y)-sd_M(z,w)]$ is, in fact, strictly positive definite on $S^m \\times S^M$.\\ If $g$ is nonconstant, then the original quadratic form is always positive unless all the $c_\\mu$ are zero.\\eop\n\nTo close the paper, we go the other way around, seeking positive definiteness on a single sphere from positive definiteness on a product of spheres.\\ Two results in that direction are as follows.\n\n\\begin{prop} \\label{restric} If $f$ is the continuous and isotropic part of a (strictly) positive definite kernel on $S^{m}\\times S^M$, then $t  \\to f(t,1)$ and $s \\to f(1,s)$ are the isotropic parts of (strictly) positive definite kernels on $S^m$ and $S^M$ respectively.\n\\end{prop}\n\n\\begin{prop} \\label{restric1} If $f$ is the continuous and isotropic part of a $DC$-strictly positive definite kernel on $S^{m}\\times S^M$, then $t  \\to f(t,t)$ is the isotropic part of a strictly positive definite kernel\n on $S^{m\\wedge M}$, in which $m\\wedge M=\\min\\{m,M\\}$.\n\\end{prop}\n\nWe intend to provide solutions for some of the problems mentioned above in a subsequent paper.\\ For now, we conclude this one mentioning a few relevant references that deals with similar questions on a single sphere: \\cite{chen, cheney,gneiting,mene,ron,sun}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a hyperconnected world, scientific and technological advances have led to an overwhelming amount of user data collected and processed by hundreds of companies over public networks. Companies mine this data to provide personalized services. These technologies have, however, come with the price of an extensive loss of privacy in society. Depending on their resources, adversaries can infer critical (private) information about system operations from public data available on the internet and\/or unsecured servers and networks. This is why researchers from different fields (e.g., computer science, information theory, and control theory) have been attracted to the broad research area of privacy and security of Cyber-Physical Systems (CPSs), \\cite{Farokhi1}\\nocite{Farokhi2}\\nocite{Pappas}\\nocite{Jerome1}\\nocite{Takashi_1}\\nocite{Takashi_3}-\\nocite{chaper_privacy_chaos}\\hspace{-.1mm}\\cite{Carlos_Iman1}.\\\\\nIn most engineering applications, information about the state of systems, say $X$, is obtained through sensor measurements and then sent to a remote station through communication networks for signal processing and decision-making purposes. If the communication network is public\/unsecured and\/or the remote station is untrustworthy, adversaries might access and estimate the system state. \\\\\nA common technique to avoid an accurate estimation is the use of additive random vectors to distort disclosed data. In the context of privacy of databases, a popular approach is differential privacy \\cite{Jerome1,Dwork}, where random noise is added to the response of queries so that private information stored in the database cannot be inferred. In general, if the data to be kept private follows continuous probability distributions, the problem of finding the optimal additive noise to maximize privacy is hard to solve. This problem has been addressed by assuming the data to be kept private is deterministic \\cite{Farokhi1,SORIA,Geng}. However, in a Cyber-Physical-Systems context, the inherent system dynamics and unavoidable system and sensor noise lead to stochastic non-stationary data and thus, existing tools do not fit this setting. More recently, the authors in \\cite{Carlos_Iman1,murguia2021privacy} have proposed a framework for synthesizing optimal transition probabilities to maximize privacy for a class of quantized CPSs with discrete multivariate probability distributions. Even though this framework is general and leads to distorting mechanisms with arbitrary distributions, the computational complexity induced by exploring all the possible transition probabilities from private to the disclosed distorted data is high and increases exponentially with the alphabet of the private data. In \\cite{Farokhi1}, the authors neglect quantization and work directly with dynamical systems driven by continuous (Gaussian) disturbances. They prove that, in the unconstrained additive noise case (so distortion is not considered), the optimal noise distribution minimizing the Fisher information (their privacy metric) is Gaussian. This observation has also been made in \\cite{Cedric} where mutual information is used as privacy metric.\\\\\nMotivated by these results, in this manuscript, we present an optimization-based framework for synthesising privacy-preserving Gaussian mechanisms that maximize privacy but keep distortion bounded. We use additive Gaussian dependent vectors as distorting mechanisms to maximize privacy. We pass sensor data and input signals through these distorting mechanisms before transmission and send the distorted data to the remote station instead. These mechanisms consist of a coordinate transformation and additive dependent Gaussian vectors that are designed to hide (as much as possible) the private parts of the state $S$ -- a desired private output modeled as some linear function of the system state, $S=DX$, for some deterministic matrix $D$.\\\\\n Note, however, that it is not desired to overly distort the original data. When designing the additive Gaussian vectors, we need to take into account the trade-off between \\emph{privacy} and \\emph{distortion}. As \\emph{distortion metric}, we use a general \\emph{weighted mean squared error} between the original and distorted data. Weighting matrices are used to fine tune the desired distortion at different channels and\/or to model different applications of the distorted data at the remote station. In this manuscript, we follow an information-theoretic approach to privacy. As \\emph{privacy metric}, we propose a combination of \\emph{mutual information} and \\emph{entropy} \\cite{Cover} between disclosed and private data. In particular, we aim at minimizing the mutual information $I[S;Z]$, between the private output $S$ and the disclosed randomized sensor data $Z$, while maximizing the entropy $h[H]$ of an additive Gaussian vector $H$ we use to distort input signals, over a finite time window, for \\emph{desired levels of distortion} -- how different actual and distorted data are allowed to be. As we prove in this manuscript, we can cast the problem of finding the optimal Gaussian distributions and change of coordinates as a constrained convex optimization problem.\\\\\n\\textbf{Notation:} The symbol $\\Real$ stands for the real numbers, $\\Real_{>0}$($\\Real_{\\geq 0}$) denotes the set of positive (non-negative) real numbers. The symbol $\\Nat$ stands for the set of natural numbers. The Euclidian norm in $\\Real^n$ is denoted by $||X||$, $||X||^2=X^{\\top}X$, where $^{\\top}$ denotes transposition. The $n \\times n$ identity matrix is denoted by $I_n$ or simply $I$ if $n$ is clear from the context. Similarly, $n \\times m$ matrices composed of only ones and only zeros are denoted by $\\mathbf{1}_{n \\times m}$ and $\\mathbf{0}_{n \\times m}$, respectively, or simply $\\mathbf{1}$ and $\\mathbf{0}$ when their dimensions are clear. For positive definite (semidefinite) matrices, we use the notation $P>0$ ($P \\geq 0$). For any two matrices $A$ and $B$, the notation $A \\otimes B$ (the Kronecker product) stands for the matrix composed of submatrices $A_{ij}B$ , where $A_{ij}$, $i,j=1,...,n$, stands for the $ij-$th entry of the $n \\times n$ matrix $A$. The notation $X \\sim \\mathcal{N}[\\mu,\\Sigma^X]$ means that $X \\in \\Real^{n}$ is a normally distributed random vector with mean $E[X] = \\mu \\in \\Real^{n}$ and covariance matrix $E[(X-\\mu)(X-\\mu)^T] = \\Sigma^X \\in \\Real^{n \\times n}$, where $E[a]$ denotes the expected value of the random vector $a$. Finite sequences of vectors are written as $X^K := (X(1)^{\\top},\\ldots,X(K)^{\\top})^{\\top} \\in \\Real^{Kn}$, $X(i) \\in \\Real^{n}$, $i=1,...,K$, and $n,K \\in \\Nat$. To avoid confusion, we denote powers of matrices as $(A)^{K} = A \\cdots A$ ($K$ times) for $K > 0$, $(A)^{0} = I$, and $(A)^{K} = \\mathbf{0}$ for $K < 0$. The operators $\\log[\\cdot]$, $\\det[\\cdot]$, and $\\text{tr}[\\cdot]$ stand for logarithm base two, determinant, and trace, respectively.\n\n\n\\section{Preliminaries}\nIn this section, we present some definitions and preliminary results needed for the subsequent sections.\n\n\\begin{definition} [MMSE Estimator \\cite{huemer2017component}] \\label{linearMMSE}\nLet $X$ and $Y$ be two jointly Gaussian random vectors. The Minimum Mean Square Error (MMSE) estimate of $X$ given $Y$ is given by\\emph{:}\n\\begin{eqnarray}\n\\hat X = \\Sigma ^{XY} {\\Sigma^Y}^{-1} \\left(Y - E(Y) \\right) + E(X),\n\\end{eqnarray}\nwhere $\\Sigma ^{XY}$ denotes the cross-covariance matrix between $X$ and $Y$ and $\\Sigma^Y$ is the covariance matrix of $Y$.\n\\end{definition}\n\n\\begin{definition}[Differential Entropy \\cite{Cover}]\\label{entropy}\nLet $S \\in \\Real^n$ with $S \\sim \\mathcal{N}[\\mu^S,\\Sigma^S]$. Its differential entropy, $h(S)$, can be written in terms of its covariance matrix as\\emph{:}\n\\begin{eqnarray}\nh[S] = \\frac{1}{2}\\log \\det \\left( \\Sigma^S \\right) + \\frac{n}{2} + \\frac{n}{2}\\log(2\\pi). \\label{entropyCov}\n\\end{eqnarray}\n\\end{definition}\nEntropy is a measure of the average uncertainty in a random vector. We use base two $\\log(\\cdot)$, so entropy is given in bits.\n\n\\begin{definition}[Mutual Information \\cite{Cover}]\\label{mutual_info}\nLet $S$ and $Z$ be two jointly distributed continuous random vectors with joint entropy $h[S,Z]$ and marginal entropies $h[S]$ and $h[Z]$. Their mutual information, $I[S;Z]$, is given as\\emph{:}\n\\begin{eqnarray}\nI[S;Z] = h[S] + h[Z] - h[S,Z].\\label{mutinf}\n\\end{eqnarray}\n\\end{definition}\nMutual information between two jointly distributed vectors is a measure of the statistical dependence between them.\n\n\\section{Problem Formulation}\n\\subsection{System Description}\nWe consider discrete-time stochastic systems of the form:\n\\begin{eqnarray}\\label{eq1}\n\\left\\{ \\begin{split}\nX(k+1) &= AX(k) + BU(k) +  T(k),\\\\\nY(k) &= CX(k) + W(k),\\\\\nS(k) &= DX(k),\n\\end{split} \\right.\n\\end{eqnarray}\nwith time-index $k \\in \\Nat$, state $X \\in {\\mathbb{R}^{{n_x}}}$, measurable output $Y \\in {\\mathbb{R}^{{n_y}}}$, \\emph{known} input $U \\in {\\mathbb{R}^{{n_u}}}$, private performance output $S \\in {\\mathbb{R}^{{n_s}}}$, and matrices $(A,B,C,D)$ of appropriate dimensions, ${n_x}, {n_y}, {n_u}, {n_s} \\in \\mathbb{N}$. Matrix $D$ is full row rank. The state perturbation $T$ and the output perturbation $W$ are multivariate i.i.d. Gaussian processes with zero mean and covariance matrices ${\\Sigma ^T}>0$ and ${\\Sigma ^W}>0$, respectively. The initial state $X(1)$ is assumed to be a Gaussian random vector with $E[X(1)]=\\mu^X_1 \\in \\mathbb{R}^{n_x}$ and covariance matrix $\\Sigma^X_1 = E[(X(1)-\\mu^X_1)(X(1)-\\mu^X_1)^{\\top}] \\in \\mathbb{R}^{n_x \\times n_x}$, $\\Sigma^X_1 > 0$. Processes $T$ and $W$ and the initial condition $X(1)$ are mutually independent. We assume that matrices (vectors) $(A,B,C,D,\\Sigma^X_1,\\mu^X_1,\\Sigma^T,\\Sigma^W)$ and the input signal $U\\left( k \\right)$ are known for all $k$.\\\\\nWe aim to prevent adversaries from estimating the private output $S(k)$, accurately. To this end, we randomize measurements $Y(k)$ and input signals $U(k)$ before transmission and send the corrupted data to the remote station instead. The idea is to randomize $Y(k)$ and $U(k)$ as\n\\begin{equation}\\label{eq2}\n\\left\\{\n\\begin{array}{ll}\n  Z(k) = G(k)Y(k) + V(k),\\\\[2mm]\n  R(k) = U(k) + H(k),\n\\end{array}\n\\right.\n\\end{equation}\nfor some time-varying  transformation $G(k) \\in {\\mathbb{R}^{{n_y} \\times {n_y}}}$ and dependent Gaussian processes, $V(k) \\sim \\mathcal{N}[\\mathbf{0},\\Sigma^V(k)]$ and $H(k) \\sim \\mathcal{N}[\\mathbf{0},\\Sigma^H(k)]$.\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{1f}\n    \\caption{System configuration.}\n    \\label{fig2}\n\\end{figure*}\nThe randomized vectors $Z(k)$ and $R(k)$ are transmitted over an unsecured communication network to a remote station, see Figure \\ref{fig2}. We seek to synthesize the sequences $G(k)$, $\\Sigma^V(k)$, and $\\Sigma^H(k)$, $k \\in \\mathcal{K}:= \\{1,\\ldots,K\\}$, to make inference of the sequence of private outputs, $S(k)$, as `hard' as possible from the disclosed data, $(Z(k),R(k))$. In what follows, we introduce the adversarial model we seek to defend against.\n\n\\subsection{Adversarial Capabilities}\nWe consider worst-case adversaries that eavesdrop data at the communication network and\/or the remote station. They do not only have access to all distorted sensor measurements $Z(k)$ and distorted input signal $R(k)$, but also have prior knowledge of the dynamics and the stochastic properties of the system, i.e., matrices  $(A,B,C,D,\\Sigma^X_1,\\mu^X_1,\\Sigma^T,\\Sigma^W)$ are known by the adversary. Moreover, the adversary also knows the means and covariance matrices $(\\mu^Z(k),\\mu^R(k),\\Sigma^Z(k),\\Sigma^R(k))$ as these can be estimated from the disclosed data $(Z(k),R(k))$. We assume that the adversary uses a linear MMSE estimator (see Definition 1) to reconstruct $S(k)$, which, for jointly Gaussian vectors, produces the best estimation performance among all unbiased estimators \\cite{huemer2017component}. In practice, actual adversaries would typically not have all the capabilities that we assume here. However, if we maximize privacy under such worst-case adversaries, we ensure that adversaries with less capabilities perform even worse (or equal at most).\n\n\n\\subsection{Metrics and Problem Formulation}\nFor a given time horizon $K \\in \\mathbb{N}$, the aim of our privacy scheme is to make inference of the sequence of private vectors, ${S^K} = (S(1)^\\top,...,S(K)^\\top)^\\top$, from the distorted disclosed sequences, ${Z^K} = (Z(1)^\\top,...,Z(K)^\\top)^\\top$ and \\linebreak ${R^{K}} = (R(1)^\\top,...,R(K)^\\top)^\\top$, as hard as possible without distorting $(Y^K,U^{K})$, excessively. That is, we do not want to make $(Y^K,U^{K})$ and $(Z^K,R^{K})$ overly different. Hence, when designing the distorting variables $(G(k),\\Sigma^V(k),\\Sigma^H(k))$, we need to consider the \\emph{trade-off between privacy and distortion}.\\\\\nAs distortion metric, we use the weighted mean squared errors between the original and distorted data, i.e., $E[||{W_Y}(Z^K - Y^K)||^2]$ and $E[||{W_U}(R^K - U^K)||^2]$, for some given weighting matrices of appropiate dimensions. Matrices $W_Y,W_U$ are used to fine-tune the desired distortion at different channels and\/or to model different applications of the distorted data at the remote station. Arguably, for the class of linear systems considered in this manuscript, and most applications at the remote station (for this class), performance degradation induced by the privacy mechanism can be written (or upper bounded) in terms of the proposed weighted mean squared errors.\\\\\nAn intuitive candidate to use as privacy metric is the mutual information between private and disclosed data, i.e., $I[S^K;Z^K,R^{K}]$. However, because the input sequence $U^{K}$ is deterministic and $V(k)$ and $H(k)$ are independent, it is easy to verify that $I[S^K;Z^K,R^{K}] = I[S^K;Z^K]$. That is, in the proposed setting, the randomized input data $R^{K}$ does not affect $I[S^K;Z^K,R^{K}]$ at all. To overcome this obstacle, we add to $I[S^K;Z^K]$ the negative differential entropy, $h[U^{K}-R^{K}] = h[H^{K}]$, to capture the uncertainty between original and disclosed input data. That is, we propose $I[S^K;Z^K] - h[H^{K}]$ as privacy metric.\\\\\nSummarizing the above discussion, we aim at minimizing $I[S^K;Z^K] - h[H^{K}]$ subject to the weighted second moment constraints $E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y$ and $E[||{W_U}(R^K - U^K)||^2] \\leq \\epsilon_U$, for some desired maximum distortion levels ${\\epsilon _Y} \\in {\\mathbb{R}_{>0}}$ and ${\\epsilon _U} \\in {\\mathbb{R}_{>0}}$ and weights ${W_Y} \\in {\\mathbb{R}^{{n_y} \\times {n_y}}}$ and ${W_U} \\in {\\mathbb{R}^{{n_u} \\times {n_u}}}$, by designing $G(k)$, ${\\Sigma^V(k)}$, and ${\\Sigma^H(k)}$ of the distorting mechanisms \\eqref{eq2}. In what follows, we formally present the optimization problem we seek to address.\n\n\\textbf{Problem 1} Given the system dynamics \\eqref{eq1}, time horizon $K\\in \\mathbb{N}$, desired maximum distortion levels ${\\epsilon _Y} \\in {\\mathbb{R}_{>0}}$ and ${\\epsilon _U} \\in {\\mathbb{R}_{>0}}$, weighting matrices ${W_Y} \\in {\\mathbb{R}^{{n_y} \\times {n_y}}}$ and ${W_U} \\in {\\mathbb{R}^{{n_u} \\times {n_u}}}$, private output sequence $S\\left( k \\right)$, $k \\in \\mathcal{K}={1,...,K}$, and the distorting mechanism \\eqref{eq2}, find the distorting sequences, $G(k)$, ${\\Sigma^V(k)}$, and ${\\Sigma^H(k)}$, solution of the following optimization problem:\n\\begin{equation} \\label{problem1}\n\\left\\{\\begin{aligned}\n\t&\\min_{G(k), {\\Sigma^V(k)}, {\\Sigma^H(k)}, k \\in \\mathcal{K}}\\ I[S^K;Z^K] - h[{H}^{K}],\\\\[1mm]\n    &\\hspace{4mm}\\text{s.t. } \\left\\{\\begin{aligned}\n    &E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y ,\\\\\n    &E[||{W_U}(R^K - U^K)||^2] \\leq \\epsilon_U\n    \\end{aligned}\\right. \\\\[1mm]\n    &\\hspace{12mm}\\text{and } (V^K,H^K,Y^K) \\text{ mutually independent}.\n\\end{aligned}\\right.\n\\end{equation}\n\n\\section{Solution to Problem 1}\nTo solve Problem 1, we first need to write the cost function and constraints in terms of the design variables.\n\n\\subsection{Cost Function: Formulation and Convexity}\nThe differential entropy $h[{H}^{K}]$ is fully characterized by the covariance matrix of ${H}^{K}$, $\\Sigma^H_K := E[H_K H_K^\\top]$. Hence, by Definition 2, $h[{H}^{K}]$ is given by\n\\begin{equation}\\label{hUprim}\nh[{H}^{K}] = \\frac{1}{2}\\log \\det \\left( \\Sigma^H_K \\right) + \\frac{Kn_u}{2} + \\frac{Kn_u}{2}\\log(2\\pi).\n\\end{equation}\n\nThe mutual information $I\\left[ {{S^K};{Z^K}} \\right]$ can be written in terms of differential entropies as $I[S^K;Z^K] = h[S^K] + h[Z^K] - h[S^K,Z^K]$, see Definition \\ref{mutual_info}. Moreover, these entropies are fully characterized by the covariance matrices of the corresponding random vectors, see Definition 2. So, to characterize $I\\left[ {{S^K};{Z^K}} \\right]$ in terms of $G(k)$ and $\\Sigma^V(k)$, $k \\in \\mathcal{K}$, we need to write the covariance matrices of $S^K$ and $Z^K$, and the joint covariance of $(S^K,Z^K)$ in terms of them. By lifting the system dynamics \\eqref{eq1} over $\\{1,\\ldots,K\\}$, we can write the stacked vector $((Z^K)^\\top,(S^K)^\\top)^\\top \\in \\mathbb{R}^{K(n_y + n_s)}$ as\n\n\\begin{align}\\label{stackedXY}\n&\\begin{bmatrix} Z^K \\\\ S^K \\end{bmatrix} = \\begin{bmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bmatrix}F_K X(1) + \\begin{bmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bmatrix} J_K T^{K-1}\\\\\n&\\hspace{15mm} + \\begin{bmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bmatrix} L_K U^{K-1} + \\begin{bmatrix} I \\\\ \\mathbf{0} \\end{bmatrix}({{\\tilde{G}}_K} W^{K}+V^{K}), \\notag\n\\end{align}\nwith stacked matrices $L_K := J_K(I_{K-1} \\otimes B)$, $\\tilde{C}_K := I_K \\otimes C$, $\\tilde{D}_K := I_K \\otimes D$, ${{\\tilde{G}}_K} := \\text{diag}\\left[ {{G_1},{G_2}, \\ldots ,{G_K}} \\right]$, and\n\\begin{equation}\\label{stackedZ}\n\\left\\{ \\begin{aligned}\n  F_K &:= \\begin{bmatrix} I & A^\\top & \\hdots & (A^\\top)^{K-1}  \\end{bmatrix}^\\top,\\\\\n  J_K &:= \\begin{bmatrix} \\mathbf{0} & \\mathbf{0} & \\mathbf{0} & \\cdots & \\mathbf{0} \\\\ I & \\mathbf{0} & \\mathbf{0} & \\cdots & \\mathbf{0} \\\\ A & I & \\mathbf{0} & \\cdots & \\mathbf{0} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\[1mm] (A)^{K-2} & (A)^{K-3} & (A)^{K-4} & \\cdots & I  \\end{bmatrix}.\n\\end{aligned} \\right.\n\\end{equation}\nLet ${{\\Sigma }^V_K} \\in \\mathbb{R}^{K n_y \\times K n_y}$ denote the \\emph{non-diagonal} covariance matrix of the stacked additive dependent vector $V^K$. Note that matrices $\\tilde{G}_K$ and ${{\\Sigma}^V_K}$ contain all the distorting variables of the output mechanism, $(G(k),\\Sigma^V(k))$, $k \\in \\mathcal{K}$. In the next lemma, we give a closed-form expression of the joint density of $(S^K,Z^K)$ (which we will need to write $I[Z^K;S^K]$ in terms of the design variables).\n\n\n\\begin{lemma}\\label{stackedDist}\n\\[ \\begin{psmallmatrix} Z^K \\\\ S^K  \\end{psmallmatrix} \\sim \\mathcal{N}\\left[\\mu^{Z,S}_K,\\Sigma^{Z,S}_K\\right],\\] with mean $\\mu^{Z,S}_K \\in \\mathbb{R}^{K(n_s + n_y)}$ and covariance matrix $\\Sigma^{Z,S}_K \\in \\mathbb{R}^{K(n_s + n_y) \\times K(n_s + n_y)}$, $\\Sigma^{Z,S}_K>0$\\emph{:}\n\\begin{eqnarray}\\label{muZS}\n\\mu^{Z,S}_K = \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}F_K \\mu^X_1 + \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix} L_K U^{K-1},\n\\end{eqnarray}\n\\begin{equation}\\label{stackeconmatrix3}\n\\begingroup\n\\renewcommand*{\\arraycolsep}{2pt}\n\\begin{aligned}\n&\\Sigma^{Z,S}_K := \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\\\ \\mathbf{0} \\end{bsmallmatrix}(I_K \\otimes \\Sigma^W) \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\\\ \\mathbf{0} \\end{bsmallmatrix}^\\top + \\begin{bsmallmatrix} I \\\\ \\mathbf{0} \\end{bsmallmatrix}{{\\Sigma }^V_K}\\begin{bsmallmatrix} I \\\\ \\mathbf{0} \\end{bsmallmatrix}^\\top\\\\[1mm]\n&+ \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix} F_K \\Sigma^X_1 F_K^\\top \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}^\\top\\\\[1mm]\n&+ \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}J_K (I_{K-1} \\otimes \\Sigma^T)J_K^\\top \\begin{bsmallmatrix} {{\\tilde{G}}_K} \\tilde{C}_K  \\\\ \\tilde{D}_K \\end{bsmallmatrix}^\\top.\n\\end{aligned}\n\\endgroup\n\\end{equation}\n\\end{lemma}\n\\emph{\\textbf{Proof}}: To simplify notation, we introduce the stacked vector ${\\Theta ^K} := {( {{{( {{Z^K}})}^\\top},{{( {{S^K}})}^\\top}} )^\\top}$. By assumption, the initial condition $X(1)$, and the processes, $T(k)$, $W(k)$, and $V(k)$, $k \\in \\mathbb{N}$, are mutually independent, and $X(1) \\sim \\mathcal{N} [ {\\mu _1^X,\\Sigma _1^X} ]$, $T\\left( k \\right) \\sim \\mathcal{N} [ {\\mathbf{0},\\Sigma^T} ]$, $W(k) \\sim \\mathcal{N} [{\\mathbf{0},\\Sigma^W} ]$, $V(k) \\sim \\mathcal{N} [ {\\mathbf{0},\\Sigma^V} ]$, for some positive definite covariance matrices $\\Sigma _1^X$, $\\Sigma^T$, $\\Sigma^W$, and $\\Sigma^V$. Then, see \\cite{Ross} for details, we have ${L_1}X\\left( 1 \\right) \\sim \\mathcal{N} [ {{L_1}\\mu _1^X,{L_1}\\Sigma _1^XL_1^\\top} ]$, ${L_2}T^{K-1} \\sim \\mathcal{N} [\\mathbf{0}, {{L_2}({I_{K - 1}} \\otimes {\\sum ^T}){L_2}^T} ]$, ${L_3}W^K \\sim \\mathcal{N} [\\mathbf{0}, {{L_3}({I_K} \\otimes {\\sum ^W}){L_3}^\\top} ]$, ${L_4}V^K \\sim \\mathcal{N} [\\mathbf{0}, {{L_4}{{\\Sigma }^V_K}{L_4}^\\top} ]$, for any deterministic matrices $L_j$,\n$j = 1, 2, 3, 4$, of appropriate dimensions. It follows that the stacked vector ${\\Theta ^K}$ given in \\eqref{stackedXY} is the sum of a deterministic vector, ${( {({{\\tilde{G}}_K} \\tilde{C}_K)^\\top,\\tilde{D}_K^T} )^\\top}{L_K}{U^{K - 1}}$, and four independent normally distributed vectors. Therefore, ${\\Theta ^K}$ follows a multivariate normal distribution with $E[ {{\\Theta ^K}} ] = \\mu^{Z,S}_K$ as in \\eqref{muZS}.\nBy inspection, using the expression of ${\\Theta ^K}$ in \\eqref{stackedXY}, mutual independence among $X(1)$, $T(k)$, $W(k)$, and $V(k)$, $k \\in N$, and the definition of $\\Sigma _1^X$, $\\Sigma _1^X = E[ {\\left( {X(1) - \\mu _1^X} \\right){{\\left( {X(1) - \\mu _1^X} \\right)}^\\top}} ]$, it can be verified that the covariance matrix of ${\\Theta ^K}$, $E[ {( {{\\Theta ^K} - E( {{\\Theta ^K}} )} ){{( {{\\Theta ^K} - E( {{\\Theta ^K}} )} )}^\\top}} ]$, is given by $\\Sigma^{Z,S}_K$ in \\eqref{stackeconmatrix3}. It remains to prove that the distribution of ${\\Theta ^K}$ is not degenerate, that is, $\\Sigma^{Z,S}_K > 0$. Note that $\\Sigma^{Z,S}_K$ in \\eqref{stackeconmatrix3} can be written as\n\\begin{align}\n\\Sigma _K^{Z,S} = \\left[ {\\begin{array}{*{20}{c}}\n\\Sigma _K^{Z} & { \\tilde{G}_K  \\tilde{C}_K Q  \\tilde{D}_K^\\top}\\\\\n{ \\tilde{D}_K Q  \\tilde{C}_K^\\top  {{\\tilde{G}}_K}^\\top}&{ \\tilde{D}_K Q  \\tilde{D}_K^\\top}\n\\end{array}} \\right],\\label{CovZS}\n\\end{align}\n\\begin{eqnarray}\n\\Sigma _K^{Z} = {\\tilde{G}_K \\tilde{C}_K Q \\tilde{C}_K^\\top  {{\\tilde{G}}_K}^\\top +  {{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K}},\n\\end{eqnarray}\nwith $Q = {F_K}\\Sigma _1^XF_K^\\top + {J_K}( {{I_{K - 1}} \\otimes {\\sum ^T}} )J_K^\\top$. A necessary condition for the block matrix $\\Sigma _K^{Z,S}$ in \\eqref{CovZS} to be positive definite is that the diagonal blocks are positive definite \\cite{Horn}.\nThe left-upper block is positive definite because by assumption ${{\\Sigma }^V_K} > 0$. The right-lower block is positive definite if $\\tilde{D}_K$ is full row rank and $Q$ is positive definite. Because $D$ is full row rank by assumption, matrix $\\tilde{D}_K = \\left( {{I_K} \\otimes D} \\right)$ is also full row rank [\\cite{Horn2}, Theorem 4.2.15].\nNote that $Q$ can be factored as follows\n\\begin{eqnarray}\\label{Q}\nQ = \\left[ {\\begin{array}{*{20}{c}}\n{{F_K}}&{{J_K}}\n\\end{array}} \\right]\\underbrace {\\left[ {\\begin{array}{*{20}{c}}\n{\\Sigma _1^X}&0\\\\\n0&{{I_{K - 1}} \\otimes {\\Sigma ^T}}\n\\end{array}} \\right]}_{Q'}\\left[ {\\begin{array}{*{20}{c}}\n{{F_K}}\\\\\n{{J_K}}\n\\end{array}} \\right],\n\\end{eqnarray}\nThat is, $Q$ is a linear transformation of the block diagonal matrix ${Q'}$ above. By inspection, it can be verified that matrix $P = [F_K \\hspace{1mm} J_K]$, see \\eqref{stackedZ}, is lower triangular with identity matrices on the diagonal; thus, $P$ is invertible. It follows that $Q = PQ'P^\\top$ is a congruence transformation of ${Q'}$ \\cite{boyd1994linear}. Hence, $Q$ is positive definite if and only if the block diagonal matrices of ${Q'}$ are positive definite \\cite{boyd1994linear}. Matrices ${\\Sigma _1^X}$ and ${\\Sigma^T}$ are positive definite by assumption (which implies ${{I_{K - 1}} \\otimes {\\Sigma ^T}}>0$), and thus we can conclude that $Q>0$, which implies ${\\tilde{D}_K Q  \\tilde{D}_K^\\top}>0$ because $\\tilde{D}_K$ is full row rank. Necessary and sufficient conditions for $\\Sigma _K^{Z,S}>0$ are ${\\tilde{D}_K Q  \\tilde{D}_K^\\top}>0$ which we have already proved) and that the Schur complement of block ${\\tilde{D}_K Q  \\tilde{D}_K^\\top}$ of $\\Sigma _K^{Z,S}$, denoted as $\\Sigma _K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top})$, is positive definite [\\cite{zhang2006schur}, Theorem 1.12]. This Schur complement is given by\n\\begin{align}\n&\\Sigma_K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top}) =\n{{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K} +\\\\ \\nonumber\n&{{\\tilde{G}}_K} \\tilde{C}_K ( {Q - Q \\tilde{D}_K^\\top ({\\tilde{D}_K Q \\tilde{D}_K^\\top} )^{-1}\\tilde{D}_K Q}) \\tilde{C}_K^\\top {{\\tilde{G}}_K}^\\top.\n\\end{align}\nSince matrix ${{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K}$ is positive definite, a sufficient condition for $\\Sigma _K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top})>0$ is\n\\begin{eqnarray}\nQ'' := ( {Q - Q \\tilde{D}_K^\\top ( {\\tilde{D}_K Q \\tilde{D}_K^\\top} )^{-1}\\tilde{D}_K Q} )>0.\n\\end{eqnarray}\nRegarding ${Q''}$ as the Schur complement of a higher dimensional matrix ${Q'''}$, we can conclude that:\n\\begin{eqnarray}\n\\begin{array}{l}\nQ'' \\ge 0 \\Leftrightarrow Q''' = \\left[ {\\begin{array}{*{20}{c}}\nQ&{Q\\tilde{D}_K^\\top}\\\\\n{\\tilde{D}_K Q}&{\\tilde{D}_K Q \\tilde{D}_K^\\top}\n\\end{array}} \\right]\\\\\n = \\left[ {\\begin{array}{*{20}{c}}\nQ\\\\\n{\\tilde{D}_K Q}\n\\end{array}} \\right]{Q^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}\nQ&{Q\\tilde{D}_K^\\top}\n\\end{array}} \\right] \\ge 0,\n\\end{array}\n\\end{eqnarray}\nwhich is trivially true because $Q^{-1}$ is positive definite since $Q>0$. Hence, ${\\tilde{D}_K Q \\tilde{D}_K^\\top}$ and $\\Sigma _K^{Z,S} \/ ({\\tilde{D}_K Q  \\tilde{D}_K^\\top})$ are both positive definite, and thus $\\Sigma _K^{Z,S}>0$.\n\\hfill $\\blacksquare$\n\nTo obtain the densities of $Z^K$ and $S^K$, we just marginalize (see \\cite{Ross}) their joint density in Lemma 1 over $S_K$ and $Z_K$.\n\n\\begin{corollary} \\label{corollary1}\n$Z^{K} \\sim \\mathcal{N} [\\mu^Z_K, \\Sigma^Z_K]$ and\n$S^{K} \\sim \\mathcal{N}[\\mu^S_K,\\Sigma^S_K]$\\emph{:}\n\\begin{align*}\n&\\mu^Z_K = {{\\tilde{G}}_K} \\tilde{C}_K F_K \\mu^X_1 + {{\\tilde{G}}_K} \\tilde{C}_K L_K U^{K-1},\\\\\n&\\mu^S_K = \\tilde{D}_KF_K \\mu^X_1 + \\tilde{D}_KL_KU^{K-1},\\\\\n&\\Sigma^Z_K = \\begin{pmatrix} I_{n_y}  \\\\ \\mathbf{0} \\end{pmatrix}^\\top \\Sigma^{Z,S}_K \\begin{pmatrix} I_{n_y} \\\\ 0  \\end{pmatrix} \\in \\mathbb{R}^{Kn_y \\times Kn_y},\\\\\n&\\Sigma^S_K = \\begin{pmatrix} \\mathbf{0}  \\\\ I_{n_s} \\end{pmatrix}^\\top \\Sigma^{Z,S}_K \\begin{pmatrix} \\mathbf{0} \\\\ I_{n_s}  \\end{pmatrix} \\in \\mathbb{R}^{Kn_s \\times Kn_s}.\n\\end{align*}\n\\end{corollary}\n\n\n\nAt this point, we have the three covariance matrices, $(\\Sigma^Z_K,\\Sigma^S_K,\\Sigma^{SZ}_K)$, that we need to compute the mutual information $I\\left[ {{S^K};{Z^K}} \\right]$. Note that these matrices are functions of our design variables $\\tilde{G}_K$ and $\\Sigma^V_K$:\n\\begin{align}\n&\\Sigma _K^{Z,S} = \\begin{pmatrix} \\Sigma _K^{Z} & (\\Sigma _K^{SZ})^\\top \\\\ \\Sigma _K^{SZ} & \\Sigma _K^{S} \\end{pmatrix},\\label{SigmaS_Z}\\\\[1mm]\n&\\Sigma_K^{Z} = {{{\\tilde{G}}_K} \\tilde{C}_K Q \\tilde{C}_K^\\top  {{\\tilde{G}}_K}^\\top +  {{\\tilde{G}}_K} (I_K \\otimes \\Sigma^W) {{\\tilde{G}}_K}^\\top + {{\\Sigma }^V_K}},\\label{eq3g}\\\\[1mm]\n&\\Sigma^S_K = { \\tilde{D}_K Q  \\tilde{D}_K^\\top},\\label{eq3h}\\\\[1mm]\n&\\Sigma _K^{SZ} = { {{\\tilde{G}}_K}  \\tilde{C}_K Q  \\tilde{D}_K^\\top},\\label{SigmaSZ}\n\\end{align}\nwith $Q$ as in \\eqref{Q} independent of $\\tilde{G}_K$ and $\\Sigma^V_K$. Before we write the cost in terms of these matrices, we notice that $\\Sigma^V_K$ only appears in the expression for $\\Sigma^Z_K$ in \\eqref{eq3g}. Moreover, given $(\\tilde{G}_K,\\Sigma^Z_K)$, matrix $\\Sigma^V_K$ is fully determined and viceversa. That is, $(\\tilde{G}_K,\\Sigma^V_K) \\rightarrow (\\tilde{G}_K,\\Sigma^Z_K)$ is an invertible transformation. Therefore, we can pose both cost and constraints in terms of either $\\Sigma^V_K$ or $\\Sigma^Z_K$. Casting the problem in terms of $\\Sigma^Z_K$ allows us to write linear distortion constraints and a convex cost function. Hereafter, we pose the problem in terms of $(\\tilde{G}_K,\\Sigma^Z_K)$. Once we have found optimal $(\\tilde{G}_K,\\Sigma^Z_K)$, we simply extract the optimal $\\Sigma^V_K$ using \\eqref{eq3g}. The first constraint that we need to enforce is that the extracted  $\\Sigma^V_K$ is always positive definite (as it is a covariance matrix). Having $\\Sigma^Z_K > \\mathbf{0}$ does not necessary imply $\\Sigma^V_K > \\mathbf{0}$. From \\eqref{eq3g}, it is easy to verify that $\\Sigma^V_K>\\mathbf{0}$ if and only if $\\Sigma_K^{Z} - {{\\tilde{G}}_K} (\\tilde{C}_K Q \\tilde{C}_K^\\top + (I_K \\otimes \\Sigma^W)) {{\\tilde{G}}_K}^\\top  > \\mathbf{0}$. Using standard Schur complement properties \\cite{zhang2006schur}, the latter nonlinear inequality can be rewritten as a higher-dimensional linear matrix inequality in $\\Sigma^Z_K$ and $\\tilde{G}_K$ as follows:\n\\begin{eqnarray}\\label{SigmaV_pos_def}\n\\left[ {\\begin{array}{*{20}{c}}\n\\Sigma _K^Z    &    {{\\tilde{G}}_K} \\\\\n{{\\tilde{G}}_K}^\\top      &    (\\tilde{C}_K Q \\tilde{C}_K^\\top + (I_K \\otimes \\Sigma^W))^{-1}\n\\end{array}} \\right] > 0.\n\\end{eqnarray}\nWe use inequality \\eqref{SigmaV_pos_def} later when we solve the complete optimization problem to enforce that the optimal $\\Sigma^Z_K$ and $\\tilde{G}_K$ lead to a positive definite $\\Sigma^V_K$.\n\nFinally, given $(\\Sigma^Z_K,\\Sigma^S_K,\\Sigma^{SZ}_K)$ in \\eqref{SigmaS_Z}-\\eqref{SigmaSZ}, by Definition 2, we can write $I[Z^K;S^K]$ as follows:\n\\begin{subequations}\n\\begin{align}\n&I[Z^K;S^K] = h\\left( S^K \\right) + h\\left( Z^K \\right) - h(Z^K,S^K)\\label{eq3a}\\\\[1mm]\n& = \\frac{1}{2} \\log \\frac{{\\det \\left( {\\Sigma _K^S} \\right)\\det \\left( {\\Sigma _K^Z} \\right)}}{{\\det \\left( {\\Sigma _K^{Z,S}} \\right)}}\\label{eq3d}\\\\\n& = \\frac{1}{2} \\log{\\det \\left( {\\Sigma _K^S} \\right)} - \\frac{1}{2} \\log{\\det  {\\left( {\\Sigma} _K^S - {{\\Sigma} _K^{SZ}}^\\top {{\\Sigma} _K^ Z}^{-1} {\\Sigma} _K^{SZ} \\right),}}\\label{eq3f}\n\\end{align}\n\\end{subequations}\nwhere \\eqref{eq3d}-\\eqref{eq3f} follow from standard determinant and logarithm formulas. In the following lemma, we prove that minimizing the cost function $I[S^K;Z^K] - h[{H}^K]$ using $({{\\tilde{G}}_K},{{\\Sigma }^Z_K},{{\\Sigma }^H_K})$ as optimization variables is equivalent to solving a convex program subject to some Linear Matrix Inequalities (LMI) constraints.\n\n\\begin{lemma}\\label{mutualinformationcov}\nMinimizing $I[S^K;Z^K] - h[{H}^K]$ is equivalent to solving the following convex program:\n\\begin{eqnarray}\n\\left\\{\\begin{aligned}\n\t&\\min_{\\Sigma^H_K, {\\Pi _K},\\Sigma_K^{Z},{{\\tilde{G}}_K}}\\\n      -\\log \\det \\left( \\Sigma^H_K \\right) - \\log{\\det \\left({\\Pi _K} \\right)} \\label{finalcost_program}\\\\[1mm]\n    &\\hspace{4mm}\\text{\\emph{s.t. }} \\Pi _K \\geq \\mathbf{0}, \\begin{bmatrix}\n\\Sigma _K^S - \\Pi _K & (\\Sigma_K^{SZ})^\\top\\\\\n\\Sigma_K^{SZ} & \\Sigma_K^Z\n\\end{bmatrix} \\geq 0.\n\\end{aligned}\\right.\n\\end{eqnarray}\n\\end{lemma}\n\\textbf{\\emph{Proof}}: {(a)} For any positive definite matrix $\\Sigma$, the function $-\\log\\det(\\Sigma)$ is convex in $\\Sigma$ \\cite{boyd2004convex}. It follows that $-h[{H}^K]$ in \\eqref{hUprim} is a convex function of $\\Sigma^H_K$. Minimizing $-h[{H}^K]$ amounts to minimizing $-\\log \\det \\left( \\Sigma^H_K \\right)$ as all other terms in \\eqref{hUprim} are constants. {(b)} Next, consider the expression for $I[Z^K;S^K]$ in \\eqref{eq3f}. Due to monotonicity of the determinant function and the fact that $\\Sigma _K^S$ is independent of the design variables, minimizing \\eqref{eq3f} is equivalent to\n\\begin{eqnarray}\n\\left\\{\\begin{aligned}\n\t&\\min_{{\\Pi _K},\\Sigma_K^{Z},{{\\tilde{G}}_K}}\\\n      - \\log{\\det \\left({\\Pi _K} \\right)} \\label{epigraphcost}\\\\[1mm]\n    &\\hspace{4mm}\\text{s.t. } \\mathbf{0} < {\\Pi _K} \\le  {\\left( {\\Sigma} _K^S - {{\\Sigma} _K^{SZ}}^\\top {{\\Sigma} _K^ Z}^{-1} {\\Sigma} _K^{SZ} \\right).} \\label{inequalityofcost}\n\\end{aligned}\\right.\n\\end{eqnarray}\n{(c)} The inequality term in \\eqref{inequalityofcost} can be rewritten using Schur complements properties  \\cite{zhang2006schur} as\n\\begin{eqnarray}\n&\\left[ {\\begin{array}{*{20}{c}}\n{\\Sigma _K^S} - {\\Pi _K} & {{\\Sigma} _K^{SZ}}^\\top\\\\\n{{\\Sigma} _K^{SZ}}&{{\\Sigma} _K^ Z}\n\\end{array}} \\right] \\ge 0 \\, {,{\\Pi _K} > 0}. \\label{finalcost}\n\\end{eqnarray}\nBecause $\\Sigma_K^{SZ}$ is a linear function of $\\tilde{G}_K$ (see \\eqref{SigmaSZ}), \\eqref{finalcost} are two LMI constraints in $\\Pi _K$ and $\\tilde{G}_K$. Combining {(a)-(c)}, we can conclude that minimizing $I[S^K;Z^K] - h[{H}^K]$ is equivalent to solving the convex program in \\eqref{finalcost_program}. \\hfill $\\blacksquare$\n\nBy Lemma \\ref{mutualinformationcov}, minimizing the cost in \\eqref{problem1} is equivalent to solving the convex program in \\eqref{finalcost_program}. Then, if the distortion metrics, $E[||{W_Y}(Z^K - Y^K)||^2]$ and $E[||{W_U}({R}^K - U^K)||^2]$, are convex in the decision variables (which is the topic of the next section), we can find optimal distorting mechanisms efficiently using off-the-shelf optimization algorithms.\n\n\\subsection{Distortion Constraints: Formulation and Convexity}\n\nWe start with the distortion in $Y^K$. By lifting the system dynamics \\eqref{eq1} over $\\{1,\\ldots,K\\}$, we can write the mean and covariance matrix of the stacked output $Y^K \\in \\mathbb{R}^{Kn_y}$ as:\n\\begin{align}\n\\mu^{Y}_K &= \\tilde{C}_K F_K \\mu^X_1 + \\tilde{C}_K L_K U^{K-1},  \\label{eq4d}\\\\[1mm]\n\\Sigma^{Y}_K &= (I_{K} \\otimes \\Sigma^W) + \\tilde{C}_K Q \\tilde{C}_K^\\top, \\label{eq4e}\n\\end{align}\nwith matrices $L_K = J_K(I_{K-1} \\otimes B)$, $\\tilde{C}_K = I_K \\otimes C$, $J_K$ in \\eqref{stackedZ}, and $Q$ in \\eqref{Q}.\n\n\\begin{lemma}\\label{constraint}\n$E[||{W_Y}(Z^K - Y^K)||^2]$ is a convex function of ${{\\Sigma }^Z_K}$ and ${{\\tilde{G}}_K}$, and can be written as follows:\n\\begin{align}\\label{distortion}\n&E[||{W_Y}(Z^K - Y^K)||^2] \\hspace{20cm} \\notag\\\\[1mm]\n&\\hspace{6mm} = \\text{\\emph{tr}}[{W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - 2\\Sigma _K^Y\\tilde{G}_K) {W_Y}]\\notag\\\\[1mm]\n& \\hspace{14mm} + {{\\mu _K^Y}^\\top} ({{\\tilde{G}}_K} - I)^\\top {W_Y}^\\top {W_Y} ({{\\tilde{G}}_K} - I)  {\\mu _K^Y}.\n\\end{align}\n\\end{lemma}\n\\textbf{\\emph{Proof}}: The expectation of the quadratic form, $E[||{W_Y}(Z^K - Y^K)||^2]$, can be written (see \\cite{seber2012linear} for details) in terms of the mean and covariance of the error $\\Delta^K := Z^K - Y^K$:\n\\begin{align}\n&E[||{W_Y}(Z^K - Y^K)||^2] = \\text{tr}[\\Sigma _K^\\Delta] + (\\mu _K^{\\Delta})^\\top \\mu _K^\\Delta \\label{eq4b}.\n\\end{align}\nwith covariance $\\Sigma _K^\\Delta := E[(\\Delta^K - \\mu _K^{\\Delta})(\\Delta^K - \\mu _K^{\\Delta})^\\top]$ and mean $\\mu _K^{\\Delta} := E[\\Delta^K]$. Given the distortion mechanism \\eqref{eq2}, we can write $Z^K = \\tilde{G}_K Y^K + V^K$; hence, $\\Delta^K = {W_Y}(({{\\tilde{G}}_K} - I) Y^K + V^K)$, and because $Y^K$ and $V^K$ are independent, $\\Sigma _K^\\Delta = {W_Y}^\\top ({{\\tilde{G}}_K} - I)^\\top \\Sigma _K^Y ({{\\tilde{G}}_K} - I) + {{\\Sigma }^V_K}) {W_Y}$ and $\\mu _K^\\Delta = {W_Y} ({{\\tilde{G}}_K} - I) \\mu _K^Y$. Note that, because $Z^K = \\tilde{G}_K Y^K + V^K$, $\\Sigma^Z_K = {{\\tilde{G}}_K}^\\top \\Sigma _K^Y {{\\tilde{G}}_K} + {{\\Sigma }^V_K}$; then, we can write $\\Sigma _K^\\Delta$ in terms of $\\Sigma^Z_K$ as  $\\Sigma _K^\\Delta = {W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - {{\\tilde{G}}_K}^\\top \\Sigma _K^Y - \\Sigma _K^Y {{\\tilde{G}}_K}) {W_Y}$. Up to this point, we have written both $\\mu _K^{\\Delta}$ and $\\Sigma _K^\\Delta$ in terms of the design variables, ${{\\Sigma }^Z_K}$ and ${{\\tilde{G}}_K}$. Using these expressions and \\eqref{eq4b}, we can conclude \\eqref{distortion}. \\hfill $\\blacksquare$\n\n\\begin{remark}\nNote that the distortion metric \\eqref{distortion} is linear in ${{\\Sigma }^Z_K}$ and quadratic in ${{\\tilde{G}}_K}$; consequently, the output distortion constraint in \\eqref{problem1}, $E[||{W_Y}(Z^K - Y^K)||^2] \\le \\epsilon_Y$, is nonlinear in the design variables. However, we can find an equivalent linear constraint using standard Schur complement properties. It can be verified (see \\emph{\\cite{zhang2006schur}}, Theorem 1.12]) that $E[||{W_Y}(Z^K - Y^K)||^2] \\le \\epsilon_Y$ in \\eqref{distortion} is equivalent to the following LMI in $\\Sigma^Z_K$ and $\\tilde{G}_K$:\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\n\\begin{bmatrix} \\theta_Y & {{\\mu _K^Y}^\\top}({{\\tilde{G}}_K} - I)^\\top {W_Y}^\\top \\\\ {W_Y} ({{\\tilde{G}}_K} - I) {\\mu _K^Y} & I \\end{bmatrix} \\ge 0,\\\\[6mm]\n\\theta_Y := \\epsilon_Y - \\text{\\emph{tr}}[{W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - 2\\Sigma _K^Y\\tilde{G}_K) {W_Y}].\n\\end{array}\n\\right.\n\\end{equation}\n\\end{remark}\n\nWe move now to the distortion in $U^K$. The input distortion metric $E[||{W_U}(R^K - U^K)||^2] = E[||{W_U}H^K||^2]$ depends on the mean $\\mu^H_K$ and covariance $\\Sigma^H_K$ of $H^K$. Note that $E[||{W_U}H^K||^2]$ is again the expected value of a quadratic form. By construction ($H^K$ is a design vector), we have $\\mu^H_K = \\mathbf{0}$. Then, $E[||{W_U}H^K||^2]$ is simply given by $E[||{W_U}H^K||^2] = \\text{tr}[{W_U}^\\top {\\Sigma }^H_K {W_U}]$ (see \\cite{seber2012linear} for details), which is already linear in the design matrix $\\Sigma^H_K$. We summarize this discussion in the following lemma.\n\n\\begin{lemma}\\label{constraintU}\n$E[||{W_U}(R^K - U^K)||^2] = E[||{W_U}H^K||^2]$ is a convex function of $\\Sigma^H_K$ and can be written as follows:\n\\begin{eqnarray}\nE[||{W_U}H^K||^2] = \\text{\\emph{tr}}[{W_U}^\\top {\\Sigma }^H_K {W_U}]. \\label{distortionU}\n\\end{eqnarray}\n\\end{lemma}\n\n\nBy Lemma \\ref{mutualinformationcov}, Lemma \\ref{constraint}, and Lemma \\ref{constraintU}, the cost function $I[S^K;Z^K] - h[{H}^K]$ and distortion constraints $E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y$ and $E[||{W_U}{H}^K||^2] \\leq \\epsilon_U$ can be written in terms of convex functions (programs) in the our optimization variables ${{\\Sigma }^Z}$, ${{\\tilde{G}}_K}$, and ${{\\Sigma }^H_K}$.\n\nIn what follows, we pose the complete nonlinear convex program to solve Problem 1.\n\n\\begin{theorem}\\label{th3}\nConsider the system dynamics \\eqref{eq1}, distorting mechanism \\eqref{eq2}, time horizon $K \\in \\mathbb{N}$, desired input and output distortion levels ${\\epsilon _U},{\\epsilon _Y} \\in {\\mathbb{R}^+}$, input and output distortion weights, ${W_Y} \\in {\\mathbb{R}^{K {n_y} \\times K {n_y}}}$ and ${W_U} \\in {\\mathbb{R}^{(K-1){n_u} \\times (K-1){n_u}}}$, covariance matrices $\\Sigma^{Z,S}_K$ and $\\Sigma _K^{S}$ and cross-covariance $\\Sigma _K^{SZ}$ \\emph{(}given in \\eqref{SigmaS_Z}-\\eqref{SigmaSZ}\\emph{)}, and the mean and covariance of $Y^K$, $\\mu^{Y}_K$ and $\\Sigma^{Y}_K$ \\emph{(}given in \\eqref{eq4d}-\\eqref{eq4e}\\emph{)}. Then, the optimization variables ${{\\Sigma }^Z}$, ${{\\tilde{G}}_K}$, and ${{\\Sigma }^H_K}$ that minimize $I[S^K;Z^K] - h[{H}^K]$ subject to distortion constraints, $E[||{W_Y}(Z^K - Y^K)||^2] \\leq \\epsilon_Y$ and $E[||W_UH^K)||^2] \\leq \\epsilon_U$, can be found by solving the convex program in \\eqref{eq:convex_optimization15}.\n\\end{theorem}\n\\emph{\\textbf{Proof:}} The expressions for the cost and constraints and convexity (linearity) of them follow from Lemma \\ref{mutualinformationcov}, Lemma \\ref{constraint}, and Lemma \\ref{constraintU}, Remark 1, and \\eqref{SigmaV_pos_def}.  \\hfill $\\blacksquare$\n\n\\begin{table}\n\\noindent\\rule{\\hsize}{1pt}\n\\begin{equation} \\label{eq:convex_optimization15}\n\\begin{aligned}\n\t&\\min_{{\\Pi _K}, \\Sigma_K^{Z}, {{\\tilde{G}}_K}, {{\\Sigma }^H_K}}  -\\log\\det [ \\Sigma^H_K ] - \\log\\det [\\Pi_K ],\\\\[1mm]\n    &\\text{ s.t. }\\left\\{\\begin{aligned}\n    &{\\Pi _K} > 0,\\\\ &\\left[ {\\begin{array}{*{20}{c}}\n{\\Sigma _K^S} - {\\Pi _K} & ( {{\\tilde{G}}_K}  \\tilde{C}_K Q  \\tilde{D}_K^\\top)^\\top\\\\\n{{\\tilde{G}}_K}  \\tilde{C}_K Q  \\tilde{D}_K^\\top &{{\\Sigma} _K^ Z}\n\\end{array}} \\right] \\ge 0, \\\\[1mm]\n&\\left[ {\\begin{array}{*{20}{c}}\n\\theta_Y & ({W_Y} ({{\\tilde{G}}_K} - I){\\mu _K^Y})^\\top\\\\\n{W_Y} ({{\\tilde{G}}_K} - I){\\mu _K^Y}&I\n\\end{array}} \\right] \\ge 0, \\\\[1mm]\n&\\theta_Y = \\epsilon_Y - \\text{tr}[{W_Y}^\\top (\\Sigma _K^Z + \\Sigma _K^Y - 2\\Sigma _K^Y\\tilde{G}_K) {W_Y}], \\\\[1mm]\n&\\epsilon_U - \\text{tr}[{W_U}^\\top {\\Sigma }^H_K {W_U}] \\ge 0,\\\\[1mm]\n&\\left[ {\\begin{array}{*{20}{c}}\n\\Sigma _K^Z    &    {{\\tilde{G}}_K} \\\\\n{{\\tilde{G}}_K}^\\top      &    (\\tilde{C}_K Q \\tilde{C}_K^\\top + (I_K \\otimes \\Sigma^W))^{-1}\n\\end{array}} \\right] > 0 \\\\\n&\\Sigma _K^H > 0. \\end{aligned}\\right.\n\\end{aligned}\n\\end{equation}\n\\noindent\\rule{\\hsize}{1pt}\n\\end{table}\n\n\n\\section{Illustrative case study}\nWe illustrate the performance of our tools through a case study of a well-stirred chemical reactor with heat exchanger. This case study has been developed over the years as a benchmark example for control systems and fault detection, see, e.g., \\cite{Wata,IET_CARLOS_JUSTIN} and references therein. The state, inputs, and output of the reactor are:\n\\begingroup\\makeatletter\\def\\f@size{9.0}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align*}\n\\left\\{\n\\begin{array}{ll}\nX(t) =  \\begin{pmatrix} C_0\\\\T_0\\\\T_w\\\\T_m \\end{pmatrix},\nU(t) =  \\begin{pmatrix} C_u\\\\T_u\\\\T_{w,u} \\end{pmatrix},\nY(t) =  T_0,\n\\end{array}\n\\right.\n\\end{align*}\\endgroup\nwhere\n\\begingroup\\makeatletter\\def\\f@size{9.0}\\check@mathfonts\n\\def\\maketag@@@#1{\\hbox{\\m@th\\normalsize\\normalfont#1}}%\n\\begin{align*}\n\\left\\{\n\\begin{array}{ll}\nC_0&: \\text{Concentration of the chemical product},\\\\\nT_0&: \\text{Temperature of the product},\\\\\nT_w&: \\text{Temperature of the jacket water of heat exchanger},\\\\\nT_m&: \\text{Coolant temperature},\\\\\nC_u&: \\text{Inlet concentration of reactant},\\\\\nT_u&: \\text{Inlet temperature},\\\\\nT_{w,u}&: \\text{Coolant water inlet temperature}.\n\\end{array}\n\\right.\n\\end{align*}\\endgroup\nWe use the discrete-time dynamics of the reactor introduced in \\cite{murguia2021privacy} with the same noise and input signals for our simulation experiments.\n\nAs private output, we use the concentration of the chemical product; then, the matrix $D$ in \\eqref{eq1} is given by the full row rank matrix $D = (1,0,0,0)$. The output of the system is the temperature of the product $T_0$, which could be monitored, e.g., for quality\/safety reasons. Then, the aim of the privacy scheme is to hide the private state $C_0$,  which is the concentration of the reactant, as much as possible without distorting output signal temperature measurements and input signals excessively.\n\\begin{figure}[!htb]\n  \\centering\n  \\includegraphics[width=3.5in]{costepsYepsUf4.eps}\n  \\caption{Evolution of the cost function for horizon $K=30$ and increasing $||\\epsilon|| = ||(\\epsilon_Y,\\epsilon_U)||$.}\\label{CostBasedEpsUEpsY}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{YZeps100f5.eps}\n  \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{YZepsinf5.eps}\n  \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{Comparison between the measurement vector $Y(k)$ and the distorted measurements $Z(k)$ for two different distortion levels, $\\epsilon_Y = 100, \\infty$.}\n\\label{YZ}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{UUpeps10f5.eps}\n  \\label{fig:sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{UUpeps1000f5.eps}\n  \\label{fig:sub-second}\n\\end{subfigure}\n\\caption{Comparison between the first element of the input signal ${U}_1(k)$ and the first element of distorted input signal ${R}_1(k)$ for two different distortion levels, $\\epsilon_U = 10, 1000$.}\n\\label{UUp}\n\\end{figure}\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{epsYinfepsU0K50f5.eps}\n  \\label{S-sub-first}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{epsY1epsU10K50f5.eps}\n  \\label{S-sub-second}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=3.5in]{epsY1epsU1000K50f5.eps}\n  \\label{S-sub-3rd}\n\\end{subfigure}\n\\caption{Comparison between the private output $S(k)$, the estimated private output without distorting $Y(k)$ and $U(k)$, ${\\hat S_{YU}}(k)$, and the estimated private output given $Z(k)$ and $R(k)$, ${\\hat S_{ZR}}(k)$, for horizon $K=50$, and three different levels of distortions, $\\epsilon_Y$ and $\\epsilon_U$.}\n\\label{estimatedS}\n\\end{figure}\nWe first show the effect of distortion levels $\\epsilon_Y$ and $\\epsilon_U$ in the cost function. Figure \\ref{CostBasedEpsUEpsY} depicts the evolution of the optimal cost $I[S^K;Z^K] - h[H^K]$ for increasing $\\epsilon = (\\epsilon_Y,\\epsilon_U)$ with time horizon $K=30$. As expected, the objective function decreases monotonically for increased maximum allowed distortion.\nFurthermore, this figure illustrates that by increasing the distortion levels, the decreasing speed of the optimal cost function will be reduced. Therefore, we can design the distortion levels such that the information leakage is minimized without distorting measurement and input signals excessively.\\\\\nThe effect of the optimal distortion mechanisms is illustrated in Figure \\ref{YZ} and Figure \\ref{UUp} for different levels of distortion, where we contrast actual and distorted data.\nNext, in Figure \\ref{estimatedS}, we depict the stacked private output $S^K$, its MMSE estimate ${{\\hat S}^K_{YU}}$ (see Definition 1) without distortion, i.e., the MMSE estimate of $S^K$ given $Y^K$ and $U^{K-1}$, and its MMSE estimate given the distorted vectors $Z^K$ and ${R}^{K-1}$, ${{\\hat S}^K_{ZR}}$. We consider horizon $K=50$, and different levels of distortion, $(\\epsilon_Y,\\epsilon_U) = \\{ (\\infty,0), (1,10), (1,1000) \\}$, to investigate the effect of each distortion level separately. $\\epsilon_Y = \\infty$ means that the optimization problem in \\eqref{eq:convex_optimization15} is solved without considering the distortion constraint between $Z^K$ and $Y^K$. As can be seen in this figure, even after distorting the measurements without the distortion constraint, the stacked private vectors ${S^K}$ can be estimated accurately by a MMSE estimator. However, by randomizing $U^K$, we can prevent an accurate estimation. By increasing the distortion level $\\epsilon_U$, the accuracy of the estimation decreases.\\\\\n\n\\section{Conclusions}\nIn this paper, for a class of stochastic dynamical systems, we have presented a detailed mathematical framework for synthesizing distorting mechanisms to minimize the information leakage induced by the use of public\/unsecured communication networks. We have proposed a class of dependent Gaussian distorting mechanisms to randomize sensor measurements and input signals before transmission to prevent adversaries from accurately estimating the private part of the system state (a performance private output).\\\\\nFurthermore, for the class of systems under study, we have fully characterized information-theoretic metrics (mutual information and differential entropy) to quantify the information between private outputs and disclosed data for a class of worst-case eavesdropping adversaries.\\\\\nFinally, given the maximum level of distortion tolerated by a particular application, we have provided tools (in terms of convex programs) to design optimal (in terms of maximizing privacy) distorting mechanisms. We have presented simulation results to illustrate the performance of our tools.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Sloan Digital Sky Survey (SDSS) photometric catalogs are an important source\nof stellar photometry, and must be understood in the context of decades of\nstellar research using different filter standards.  In this paper a transformation\nis computed between the available SDSS photometry in October 2001 and stars of the\nHK objective prism survey as provided by T. Beers, private communication.  A \ndescription of the HK survey and an earlier version of the catalog can be found at \n\\citet{psb91}.  The overlaps between the catalogs are relatively few because the \nfaint limit of the HK survey is at approximately the same magnitude as the saturation \nlimit of the SDSS.  Neither of the catalogs compared here are standard versions,\nbut were the only comparables available at the time.  Technical SDSS details can be\nfound in \\citet{getal98}; \\citet{hetal01}; \\citet{petal03}; \\citet{setal02}; and \n\\citet{yetal00}.  More recent\ntransformation equations can be obtained from \\citet{jetal05}; \\citet{kbt05}; \n\\citet{bkt05}; and \\citet{wwh05};\nand two other unpublished determinations that can be found in the documentation\nfor the SDSS DR4 at:\n\n{\\it http:\/\/www.sdss.org\/dr4\/algorithms\/sdssUBVRITransform.html }.\n\n\\section{Query Consideration and Data Reduction}\nTable 1 lists the stars that were common to the HK survey and the SDSS survey in October\n2001.  Generation of both catalogs was a work in progress at that time.\nWe selected from the HK catalog all stars whose J2000 coordinates matched\nSDSS catalog entries within:\n\\begin{eqnarray}\n\\Delta(RA)<7.\"2 (0.002^{\\circ})& \\Delta(Dec)<7.\"2\n\\end{eqnarray}\nWe rejected any matches in which the SDSS star was saturated by checking the catalog\nflags:\n\\begin{equation}\n(objFlags\\,\\, \\& \\,\\,OBJECT\\_SATUR) == 0 \n\\end{equation}\nWe use the magnitude calculated from a fit of modeled stellar profile\n(point-spread-function, or PSF magnitude) to each object. \nBecause these stars are too bright for sky noise affect the quality\nof the photometry, the use of aperture magnitudes would have made little \ndifference.\n\nAll photometry in Table 1 has been corrected for interstellar reddening\nusing the $E_{B-V}$ determined from the HK survey.  Corrections for the\nSDSS photometry are determined from the standard extinction curve of\n\\citet{ccm89}, which for SDSS filters yields:\n\\begin{eqnarray}\nA_{u^*}=5.2E_{B-V}; & A_{g^*}=3.2E_{B-V};& A_{r^*}=2.8E_{B-V}; \\nonumber\\\\\nA_{i^*}=2.1E_{B-V}; & A_{z^*}=1.5E_{B-V} \\nonumber\n\\end{eqnarray}\nWe noticed that the value of $(U-B)$ is -9.990 for some stars in the original HK list. \nThey were replaced by \"$\\cdots$\" in the Table 1. When we calculate the \ntransformation which requires $U$ band, these stars were not included.\nThree stars were removed from the original matched list because\ntheir SDSS photometry was suspicious: (RA, dec) = \n{\\bf (199.9265, 3.9129)}, {\\bf (224.7835, -0.2537)}, {\\bf (200.7934, 4.6075)}.\nTwo of the stars were close to other bright starts which may have confused the\nSDSS object deldender.\n\n\\begin{figure}\n\\plotone{proveUBV.ps}\n\\caption{Color-Color plots of UBV system against SDSS PSF magnitude.  The solid\nline is the least squares fit to the data.  The dashed line indicates the\ntransformation equation of Fukugita et al. (1996).}\n\\end{figure}\n\n\\section{Comparison of derived transformation to Fukugita et al. 1996}\n\nFigure 1 shows the color-color plots for the SDSS PSF magnitude against the HK\nsurvey U-B-V magnitudes of these stars. The coeffices in the equations, plotted as the lines, \nare derived using the {\\bf\\it Method of Least Squares}. We notice that one star, \n{\\bf (221.5996, -0.1158)} is half a magnitude brighter in SDSS filters than we expect\n(though the colors are the same). It is labeled using a bold font in Table 1. \nThere is no reason to remove it from the catalog but we ignore \nit in all fits for filter transformations.\n\n\\citet{fetal96} described SDSS photometric system and gave \nthe approximate color transformation equations from the \nJohnson-Morgan-Cousins system to the SDSS system.  A comparison of our\ntransformations to theirs is given below:\n\n\\begin{eqnarray*}\n   Fukugita's\\, paper  &  & Our Result \\nonumber\\\\\ng^*=V+0.56(B-V)-0.12 & & g^*=V+0.592(B-V)-0.102 \\nonumber\\\\\nr^*=V-0.49(B-V)+0.11 & & r^*=V-0.451(B-V)+0.082 \\nonumber\\\\\nu^*-g^*=1.38(U-B)+1.14 & & u^*-g^*=1.210(U-B)+1.103  \\nonumber\\\\\ng^*-r^*=1.05(B-V)+1.14 & & g^*-r^*=1.043(B-V)-0.185  \\nonumber\n\\end{eqnarray*}\n\nFor the blue stars from which our transformation was derived, our transformations \nare similar to Fukugita's.  The ($u^*-g^*$) transformation is the only one that\nis significantly discrepant.  This is because the\nactual $u^*$ filter response is different from the theoretical curve \nused by Fukugita et al.(1996).\n\n\\section{Inverse Transformation Equations}\n\nWe will discuss the inverse transformation from SDSS PSF magnitude to UBV system \nin this section.  Magnitudes in the $g^*$ filter and $g^*-r^*$ color are used as the \nprimary parameters in the equations since the noise in $u^*$ is typically higher. \nWe also considered other combinations of filters. \nFigures 2 and 3 show the results; the transformation is summaried\nbelow: \n\\begin{figure}\n\\plotone{tran1.ps}\n\\caption{Transformation plot from SDSS photometric system to UBV system.}\n\\end{figure}\n\n\\begin{figure}\n\\plotone{tran2.ps}\n\\caption{Transformation plot from SDSS photometric system to UBV system.}\n\\end{figure}\n\n\\begin{eqnarray*}\nU=g^*+0.883(u^*-g^*)-0.717 & & U=u^*-0.117(u^*-g^*)-0.717 \\nonumber\\\\\nB=g^*+0.348(g^*-r^*)+0.175 & & B=g^*+0.162(u^*-g^*)+0.094 \\nonumber\\\\\nV=g^*-0.561(g^*-r^*)-0.004 & & V=r^*+0.439(g^*-r^*)-0.004 \\nonumber\\\\\n(U-B)=0.754(u^*-g^*)-0.835  \\nonumber \\\\\n(B-V)=0.916(g^*-r^*)+0.187  \\nonumber\n\\end{eqnarray*}\n\n\\section{Conclusion}\n\nWe have derived transformation equations between HK catalog photometry and\nSDSS photometry.\n\n\\acknowledgments\n\nFunding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http:\/\/www.sdss.org\/.\n\nThe SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\indent A Riemannian metric $g$ on a $4$-manifold is called  hyperk\\\"ahler if its holonomy group  $Hol(g)$ is contained in $Sp(1)=SU(2)$. A closed hyperk\\\"ahler 4-manifold is diffeomorphic to either a torus or the K3 manifold, and the moduli space of all hyperk\\\"ahler metrics are described by Torelli theorems. There have been extensive recent studies on the Gromov-Hausdorff compactification of these moduli spaces, see for example \\cite{odaka2018collapsing} \\cite{sun2021collapsing}.\n\nHyperk\\\"ahler metrics in dimension 4 are the simplest models for Riemannian metrics with special holonomy. Little general existence theory is developed for the latter in dimensions greater than $4$, except for Calabi-Yau manifolds. Recently Donaldson \\cite{donaldson2018elliptic} proposes to study special holonomy metrics on manifolds with boundary and set up suitable elliptic boundary value problems. To make further progress in this direction, it is clear that we need a compactness theory.\n\nIn this paper, we study the boundary value problem for hyperk\\\"ahler 4-manifolds, which serves as the first step towards Donaldson's program.  We follow the general set-up by Fine-Lotay-Singer \\cite{fine2017space} in terms of \\emph{hyperk\\\"ahler triples}. \nA hyperk\\\"ahler triple on an oriented smooth 4-manifold $X$ is a triple of symplectic forms $\\bm{\\omega}=(\\omega_1, \\omega_2, \\omega_3)$  satisfying the following pointwise condition$$\\omega_i\\wedge\\omega_j=\\frac{1}{3}\\delta_{ij}(\\omega_1^2+\\omega_2^2+\\omega_3^2).$$\nIt is well-known that a hyperk\\\"ahler triple $\\bm\\omega$ uniquely determines a compatible hyperk\\\"ahler metric $g_{\\bm\\omega}$ such that for each $i$, $\\omega_i^2=2\\text{dvol}_{g_{\\bm\\omega}}$ and $\\omega_i$ is parallel with respect to the Levi-Civita connection. Conversely, given a hyperk\\\"ahler metric $g$ on $X$, one can choose an orientation and find a compatible hyperk\\\"ahler triple $\\bm\\omega$, which is unique up to a $SO(3)$ rotation.  \n\n\n\n\nNow let $X$ be a compact oriented  smooth 4-manifold with boundary $\\partial X$.  Note $\\partial X$ has an induced orientation defined by contracting a volume form of $X$ with an outward vector field. If $\\bm\\omega$ is a hyperk\\\"ahler triple on $X$, then its restriction to $\\partial X$  is a closed framing $\\bm\\gamma$ on $\\partial X$. The following is a natural filling problem, proposed by \\cite{fine2017space}.\n\n\\begin{question}\\label{question filling}\nWhich closed framing $\\bm\\gamma$ extends to a hyperk\\\"ahler triple on $X$?\n\\end{question}\n\nNotice a framing $\\bm\\gamma$ defines a Riemannian metric $g_{\\bm\\gamma}$ on $\\partial X$ as follows: first, there exists a unique dual coframe $\\bm\\eta=(\\eta_1,\\eta_2,\\eta_3)$ such that $\\gamma_i=\\frac{1}{2}\\delta^{ijk}\\eta_j\\wedge\\eta_k$ and such that  $\\eta_1\\wedge\\eta_2\\wedge\\eta_3$ is compatible with the orientation of $\\partial X$; then the Riemannian metric $g_{\\bm\\gamma}$ is defined by setting $\\bm\\eta$ to be orthonormal. When there is no ambiguity, we always use $\\bm\\eta$ to denote the dual coframe of $\\bm\\gamma$ defined in this way and denote the Hodge star operator of the Riemannian metric by $*_{\\bm\\gamma}=*_{\\bm\\eta}$. It is well-known that if $\\bm\\omega$ is a hyperk\\\"ahler triple, then $g_{\\bm\\omega}|_{\\partial X}=g_{\\bm\\gamma}$; more importantly that the second fundamental form of $\\partial X$ is determined intrinsically by $\\bm\\gamma$ via the  \nmatrix  $*_{\\bm\\eta}(\\eta_i\\wedge d\\eta_j)$. In particular, the mean curvature  $H_{\\bm\\gamma}$ is given by one half of the trace of this matrix, i.e.,  $H_{\\bm\\gamma}=\\frac{1}{2}*_{\\bm\\eta}(\\bm\\eta\\wedge d\\bm\\eta^T)$. \n\nThere are some previous works on Question \\ref{question filling}.    Bryant \\cite{MR2681703} studied the local ``thickening'' problem and obtained both positive and negative results.   It was shown that any real analytic closed framing on a closed oriented 3-manifold $Y$ can be extended to a hyperk\\\"ahler triple on  $Y\\times (-\\epsilon,\\epsilon)$ for some $\\epsilon>0$, and the extension is essentially unique. On the other hand, there exists a smooth closed framing on an open ball  $B^3\n\\subset \\mathbb{R}^3$ that cannot be extended to a hyperk\\\"ahler triple on $B^3\\times (-\\epsilon,\\epsilon)$ for any $\\epsilon>0$. Fine-Lotay-Singer\\cite{fine2017space} studied the local deformation theory for Question \\ref{question filling} and showed that the boundary framings must deform in certain directions. Roughly speaking, let $X=B^4$ for simplicity, suppose $\\bm\\omega$ is a hyperk\\\"ahler triple such that $\\partial X$ has positive mean curvature, and $\\bm\\omega'$ is a nearby hyperk\\\"ahler triple,\n then after moduling out diffeomorphisms of $\\partial X$, the dual coframe of $\\bm\\omega'|_{\\partial X}$ must be a small pertubation of that of $\\bm\\omega|_{\\partial X}$ in the direction of negative frenquency of the boundary Dirac operator defined by $g_{\\bm\\omega}|_{\\partial X}$.\n\n\n\nA sequence of pairs of smooth covariant tensors $(T_i^1,\\cdots, T_i^m)$ on a compact manifold $M$ with empty or nonempty boundary is said to converge in \\emph{Cheeger-Gromov} sense to $(T^1,\\cdots, T^m)$ on $M$, if there exist diffeomorphisms $f_i:M\\rightarrow M$ such that $f_i^*T_i^{1}\\rightarrow T^1,\\cdots, f_i^*T_i^m\\rightarrow T^m $ smoothly on $M$.\n\n\n\n\nOur main result is the following closedness result for Question \\ref{question filling} :\n\n\n\\begin{theorem}\\label{convergence of triples}\nLet $X$ be a compact oriented smooth 4-manifold with boundary, such that there does not exist  $C\\in H_2(X,\\mathbb{Z})$ with self intersection $C^2=-2$. Let $\\bm\\omega_i$ be a sequence of smooth hyperk\\\"ahler triples on $X$. Suppose $\\bm\\omega_i|_{\\partial X}$ converges in Cheeger-Gromov sense to a closed framing $\\bm\\gamma$ on $\\partial X$ such that $H_{\\bm\\gamma}>0$, then there exists a smooth hyperk\\\"ahler triple $\\bm\\omega$ on $X$ with $\\bm\\omega|_{\\partial X}=\\bm\\gamma$ and $\\bm\\omega_i$ converges in Cheeger-Gromov sense to $\\bm\\omega$ on $X$.\n\n\\end{theorem}\nThe proof here includes two parts: the compactness and uniqueness. The former is the main story of this paper, and the latter is a consequence of \\cite{biquard:hal-02928859} or \\cite{anderson2008unique} on unique continuation of Einstein metrics with prescribed boundary metric and second fundamental form. It is worth noting that for the compactness part, no general Riemannian convergence theory can be applied directly. The difficulty here is that we only have data on the boundary, and a priori we do not know anything near the boundary or in the interior. Specifically, we worry about the following three bad geometric behaviours:  curvature blow up, volume collapsing and boundary touching. These things are entangled, making it difficult to rule out any of them. However, we are able to separate these bad behaviours and rule them out. We will also give examples to demonstrate that the assumptions in Theorem \\ref{convergence of triples} are essential, see Remark \\ref{positive mean curvature essential} and \\ref{no -2 curve essential}.\n\n\nSuch $C$ in the assumption of Theorem \\ref{convergence of triples} is usually called a ``\\emph{$-2$ curve}'' in $X$, which appears in Kronheimer's classification of hyperk\\\"ahler ALE spaces \\cite{kronheimer1989construction} \\cite{kronheimer1989torelli}. They appear in bubble limits of volume-noncollapsed hyperk\\\"ahler manifolds. From this, one can replace the ``no $-2$ curve'' condition by an assumption on enhancements of $\\bm\\omega_i|_{\\partial X}$. Let $\\bm\\gamma$ be a closed framing on $\\partial X$. Following \\cite{Donaldson2017BoundaryVP}\\cite{donaldson2018elliptic}, an enhancement of $\\bm\\gamma$ is an equivalent class in the set of triples of  closed 2-forms on $X$ whose restrictions to $\\partial X$ are equal to $\\bm\\gamma$. The equivalence relation is defined by $\\bm\\theta\\sim \\bm\\theta+d\\bm a$, where $\\bm a$ is a triple of smooth 1-forms on $X$ vanishing on $\\partial X$. From the de Rham cohomology exact sequence of the pair $(X,\\partial X)$, \n$$H^2(X,\\partial X)\\rightarrow H^2(X)\\rightarrow H^2(\\partial X)\\rightarrow H^1(X,\\partial X),$$\nwe know $\\bm\\gamma$ has at least one enhancement if and only if each $\\gamma_i$ lies in the kernel of $H^2(\\partial X)\\rightarrow H^1(X,\\partial X)$, and we know the set of all enhancements of $\\bm\\gamma$ is an affine space over $H^2(X,\\partial X)\\otimes\\mathbb{R}^3$. Choose an enhancement of $\\bm\\gamma$ and denote it by $\\hat{\\bm\\gamma}$.  Given a 2-cycle $\\Sigma\\in H_2(X,\\mathbb{Z})$, then for any triple of closed 2-forms $\\bm\\theta\\in\\hat{\\bm\\gamma}$, $\\int_{\\Sigma}\\bm\\theta$ does not depend on the choice of $\\bm\\theta$ and we denote this invariant by $c_{\\hat{\\bm\\gamma},\\Sigma}\\in\\mathbb{R}^3$.\n\nThe proof of Theorem \\ref{convergence of triples} easily adapts to \n\\begin{theorem}\\label{convergence of triples enhancements}\nLet $X$ be a compact oriented smooth 4-manifold with boundary. Let $\\bm\\omega_i$ be a sequence of smooth hyperk\\\"ahler triples on $X$, and $\\hat{\\bm\\gamma}_i$ be the enhancement of $\\bm\\gamma_i=\\bm\\omega_i|_{\\partial X}$ where $\\bm\\omega_i$ lie in. Let $a>0$ be a positive number. Suppose for any $C\\in H_2(X,\\mathbb{Z})$ with self intersection $C^2=-2$, $|c_{\\hat\\gamma_i,C}|\\geq a$ and $\\bm\\omega_i|_{\\partial X}$ converges in Cheeger-Gromov sense to a closed framing $\\bm\\gamma$ on $\\partial X$ such that $H_{\\bm\\gamma}>0$.\nThen there exists a smooth hyperk\\\"ahler triple $\\bm\\omega$ on $X$ with $\\bm\\omega|_{\\partial X}=\\bm\\gamma$, and $\\bm\\omega_i$ converges in Cheeger-Gromov sense to $\\bm\\omega$ on $X$.\n\\end{theorem}\n\n\n\nIt is worth noting that Question \\ref{question filling} is not an elliptic boundary value problem, observed by \\cite{fine2017space}. This can also be seen from the uniqueness result of \\cite{biquard:hal-02928859} or \\cite{anderson2008unique}: the restriction of $\\bm\\omega$ to any open boundary portion determines $g_{\\bm\\omega}$ in the whole interior up to local isometries. So, it is natural to enlarge the class of closed triples of 2-forms on $X$ to obtain an elliptic boundary value problem. In \\cite{donaldson2018elliptic}, Donaldson studied the deformation theory of torsion-free $G_2$ structures on a compact oriented 7-manifold with boundary $M^7$ as follows. Suppose $\\phi_0$ is a smooth torsion-free $G_2$ structure, $\\rho_0=\\phi_0|_{\\partial M^7}$, and denote the enhancement (defined in an analogous way as before) of $\\rho_0$ where $\\phi_0$ lies in by $\\hat\\rho_0$.  Donaldson set up an elliptic boundary value problem for the torsion-free equation. So in particular, if the kernel space at $\\phi_0$ is trivial, then for any small closed 3-form $\\theta$ on $X$,  $\\hat\\rho_0+\\theta|_{\\partial X}$ contains a unique torsion-free $G_2$ structure that is close to $\\phi_0+\\theta$ after gauge fixing. This phenomenon is quite different from the hyperk\\\"ahler case, since in that case we cannot deform the boundary framing arbitrarily to extend it to a hyperk\\\"ahler triple, as we discussed before. \n \n\n\n \nIt is well-known that $G_2$ structures have a reduction to dimension 4. Consider $X^4\\times T^3$. A triple of two forms $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ on $X^4$  defines a 3-form on $X^4\\times T^3$ by\n\\begin{equation}\\label{g2 and hypersymplectic}\n \\phi=dt^1\\wedge dt^2\\wedge dt^3-\\omega_1\\wedge dt^1-\\omega_2\\wedge dt^2-\\omega_3\\wedge dt^3.\n\\end{equation}\n The triple $\\bm\\omega$ is called torsion-free hypersymplectic if $\\phi$ is a torsion-free $G_2$ structure. Locally, this is a weaker condition than being hyperk\\\"ahler, see examples in \\cite{Donaldson2017BoundaryVP} or \\cite{fine2020report}. Donaldson observed in  \\cite{Donaldson2017BoundaryVP} that the boundary value problem for torsion-free $G_2$ structures can also be reduced to dimension 4. So, a compactness result is helpful to solve this dimension reduced boundary value problem. \n\nSimilar to the hyperk\\\"ahler case, a torsion-free hypersymplectic triple $\\bm\\omega$ defines a Riemannian metric $g_{\\bm\\omega}$ and a positive definite $SL(3,\\mathbb R)$-valued function $\\bm Q=(Q_{ij})$ such that \n$$\\omega_i\\wedge \\omega_j=2Q_{ij}\\text{dvol}_{g_{\\bm\\omega}}$$\nWe denote $\\bm Q'$ the restriction of $\\bm Q$ to $\\partial X$. When there is ambiguity, we use notations $\\bm Q_{\\bm\\omega}$, $\\bm Q'_{\\bm\\omega}$ to denote their dependence on $\\bm\\omega$. One can show that the mean curvature of $\\partial X$ has an explicit expression in terms of $\\bm\\gamma$, $\\bm Q'$, and we denote this explicit expression by $H_{\\bm\\gamma,\\bm Q'}$. Note that on $\\partial X$,  $\\bm\\gamma,\\bm Q'$ are subject to the constraints $d\\bm\\gamma=0,  d(\\bm\\gamma(\\bm Q')^{-1})=0$. \n\n\nWe have the following analogue of Theorem \\ref{convergence of triples enhancements}:\n\n\n\n\n\\begin{theorem}\\label{convergence of hypersymplectic triples}\nLet $X$ be a compact oriented smooth 4-manifold with boundary. Let $\\bm\\omega_i$ be a sequence of smooth torsion-free hypersymplectic triples on $X$, and $\\hat{\\bm\\gamma}_i$ be the enhancement of $\\bm\\gamma_i=\\bm\\omega_i|_{\\partial X}$ where $\\bm\\omega_i$ lie in.  Let $a>0$ be a positive number. Suppose for any $C\\in H_2(X,\\mathbb{Z})$ with self intersection $C^2=-2$, $|c_{\\hat\\gamma_i,C}|\\geq a$, and $(\\bm\\gamma_i,\\bm Q_i')$  converges in Cheeger-Gromov sense to some pair $(\\bm\\gamma,\\bm Q')$ on $\\partial X$, such that $\\bm\\gamma$ is a framing, $\\bm Q'$ is positive definite and $H_{\\bm\\gamma,\\bm Q'}>0$. Then there exists a smooth torsion-free hypersymplectic triple $\\bm\\omega$ on $X$ with $\\bm\\omega|_{\\partial X}=\\bm\\gamma$, $\\bm Q_{\\bm\\omega}'=\\bm Q'$, and $\\bm\\omega_i$\nconverges in Cheeger-Gromov sense to $\\bm\\omega$.\n\n\\end{theorem}\n \nNote that Theorem \\ref{convergence of hypersymplectic triples} includes the previous two versions.\n\nThis paper is organized as follows: In Section \\ref{section triples}, we discuss basics of hyperk\\\"ahler triples on manifolds with boundary. In Section \\ref{section Riemannian geometry}, we discuss Riemannian geometry for manifolds with boundary and Riemannian convergence theory. In Section \\ref{section main proof}, we prove Theorem \\ref{convergence of triples}  and Theorem \\ref{convergence of triples enhancements} and give some remarks about the proofs. In Section \\ref{section torsion-free triples}, we discuss some basics for torsion-free hypersymplectic triples and prove Theorem \\ref{convergence of hypersymplectic triples}.\n\n\n\\textbf{Notations.}\n$$\\mathbb{R}_+^n=\\{x\\in\\mathbb{R}^n:x^n\\geq 0\\},$$\n $$B_r=\\{x\\in\\mathbb{R}^n: |x|<r\\},$$\n $$B_r^+=B_r\\cap \\mathbb{R}_+^n,$$\n $$\\tilde B_r=B_r^+\\cap \\partial\\mathbb{R}_+^n,$$\n$$\\partial \\tilde{B}_r=\\{x\\in \\mathbb{R}^n:|x|=r,x^n=0\\},$$\n$$\\partial^+ {B}_r^+=\\{x\\in \\mathbb{R}^n:|x|=r,x^n>0\\}.$$\n\n\\textbf{Acknowledgements.} The author is very grateful to his thesis advisor Song Sun for suggesting the problem, constant support and many inspiring discussions. I thank Antoine Song and Chengjian Yao for some useful comments. I thank Simon Donaldson and Jason Lotay for their interest in this work. \n\nThe author is supported by Simons Collaboration Grant on Special Holonomy in Geometry, Analysis, and Physics (488633, S.S.).\n\n\\section{Hyperk\\\"ahler triples and closed framings}\\label{section triples}\n\nThe discussions in this section are well-known.\n\n\\subsection{Pointwise theory}\n Let $V$ be an oriented 4-dimensional vector space, and ${\\bm\\omega}=(\\omega_1,\\omega_2,\\omega_3)\\in\\Lambda^2(V^*)\\otimes \\mathbb{R}^3$. Suppose $\\bm\\omega$ is a \\emph{definite} triple, i.e., \n$\\omega_i$ spans a maximum positive subspace of $\\Lambda^2(V^*)$ with respect to the wedge product, then $\\omega_i$ defines a unique conformal structure on $V$ by making each $\\omega_i$ self dual. Fix a volume form $\\mu_0$ on $V$ that defines the orientation of $V$, write $\\omega_i\\wedge\\omega_j=2q_{ij}\\mu_0$, we define a matrix $\\bm Q$ associated to the definite triple $\\bm\\omega$ by $Q_{ij}=\\frac{q_{ij}}{\\det(q_{ij})^\\frac{1}{3}}$, which does not depend on the choice of $\\mu_0$. We use $\\bm Q^{-1}= (Q^{ij})$ to denote the inverse matrix of $\\bm Q$. If we write $$\\omega_i\\wedge\\omega_j=2 Q_{ij}\\mu,$$ then $\\mu$ is a volume form intrinsically defined by $\\bm\\omega$. We define a unique metric $\\langle,\\rangle_{\\bm\\omega}$ on $V$ in the conformal structure by making $\\mu$ the volume form. Explicitly, $$\\langle u,v\\rangle_{\\bm\\omega}=\\frac{1}{6}\\sum_{i,j,k=1}^3 \\delta^{ijk}\\frac{\\iota_u \\omega_i\\wedge\\iota_v\\omega_j\\wedge\\omega_k}{\\mu}.$$\n So $$\\langle u,u \\rangle_{\\bm\\omega}=\\frac{\\iota_u\\omega_1\\wedge\\iota_v\\omega_2\\wedge\\omega_3}{\\mu}.$$\nDenote $*_{\\bm\\omega}$ the Hodge star operator defined by this metric.\n\nLet $W$ be an orientated 3-dimensional  vector space, and ${\\bm\\gamma}=(\\gamma_1,\\gamma_2,\\gamma_3)$ be a framing on $W$, i.e., a basis for $\\Lambda^2W^*$. Then by elementary linear algerba, there exists a coframe  ${\\bm\\eta}=(\\eta_1,\\eta_2,\\eta_3)$ such that\n\n\\begin{equation}\\label{framing and coframe}\n\\gamma_i=\\frac{1}{2}\\delta^{ijk}\\eta_j\\wedge\\eta_k.\n\\end{equation}\nSuch $\\bm\\eta$ is uniquely determined up to a sign and we choose $\\bm\\eta$ such that $\\eta_1\\wedge\\eta_2\\wedge\\eta_3$ defines the orientation of $W$ and we denote this volume form by $\\text{vol}_{\\bm\\gamma}$. There is a unique metric on $W$, denoted by $\\langle,\\rangle_{\\bm\\gamma}$, that makes $\\bm\\eta$ an orthonormal coframe. Denote $*_{\\bm\\gamma}=*_{\\bm\\eta}$ by the Hodge star operator of $\\langle,\\rangle_{\\bm\\gamma}$. Let $e_i\\in W$ be the dual vector of $\\eta_i$, so $\\eta_i(e_j)=\\delta_{ij}$.  Then $\\bm e=(e_1,e_2,e_3)$ is a frame of $W$. Conversely, given a coframe $\\bm\\eta=(\\eta_1,\\eta_2,\\eta_3)$ on $W$ compatible with the orientation, one can define a framing $\\bm\\gamma=(\\gamma_1,\\gamma_2,\\gamma_3)$ via (\\ref{framing and coframe}), and a volume form $\\text{vol}_{\\bm\\eta}=\\text{vol}_{\\bm\\gamma}$, a metric $\\langle,\\rangle_{\\bm\\eta}=\\langle,\\rangle_{\\bm\\gamma}$, a Hodge star opertaor $*_{\\bm\\eta}=*_{\\bm\\gamma}$. \n\n\nNow if $W\\subset V$ is a 3-dimensional subspace, $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ is a definite triple on $V$,   and $\\bm\\gamma=(\\gamma_1,\\gamma_2,\\gamma_3)$ is the restriction of $\\bm\\omega$ to $W$.  Let $\\langle,\\rangle_{W}$ be the restriction of the metric $\\langle,\\rangle_{\\bm\\omega}$ on $W$, which defines a volume form $\\text{vol}_W$ and a Hodge star operator $*_W$  compatible with the orientation of $W$.  Since $\\omega_i$ are self-dual, we can write $$\\omega_i=\\nu^*\\wedge *_W\\gamma_i+\\gamma_i,$$ where $\\nu^*=*_{\\bm\\omega}\\text{vol}_W$, then we have $$\\omega_i\\wedge\\omega_j=2\\nu^*\\wedge*_W\\gamma_i\\wedge\\gamma_j=2\\langle \\gamma_i,\\gamma_j\\rangle_{W}\\nu^*\\wedge \\text{vol}_{W}=2\\langle\\gamma_i,\\gamma_j\\rangle_W\\mu.$$ Hence on $W$, $$Q_{ij}=\\langle\\gamma_i,\\gamma_j\\rangle_W,$$ Since $\\det(\\bm Q)=1$, we have $\\text{vol}_{\\bm\\gamma}=\\text{vol}_{W}$. If furthermore we assume $Q_{ij}=\\delta_{ij}$, then $\\langle\\gamma_i,\\gamma_j\\rangle_{W}=\\delta_{ij}=\\langle \\gamma_i,\\gamma_j\\rangle_{\\bm\\gamma}$. In this case, $\\langle,\\rangle_{\\bm\\gamma}=\\langle,\\rangle_W$ as an inner product on $W=\\Lambda^1 W$, so in particular $*_W=*_{\\bm\\gamma}$. \n\n\n\n\n\n\n\n\\subsection{Local theory}\nNow we move our pointwise discussions to manifolds. Let $X$ be a oriented 4-manifold with boundary, $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ be a smooth section in $\\Gamma(X, \\Lambda^2T^*X\\otimes\\mathbb{R}^3)$ such that it is a definite triple pointwise, and $\\bm\\gamma=(\\gamma_1,\\gamma_2,\\gamma_3) $ be its restriction to $\\partial X$. By the discussions above, $\\bm\\omega$ defines a matrix valued function $\\bm Q=(Q_{ij})$, a volume form $\\mu$, and a Riemannian metric $g_{\\bm\\omega}$ which equals to  $\\langle,\\rangle_{\\bm\\omega}$ on the tangent spaces at each point. Similarly, $\\bm\\gamma$ defines $\\bm\\eta=(\\eta_1,\\eta_2,\\eta_3)\\in \\Omega^1(\\partial X)\\otimes\\mathbb{R}^3,\\bm e=(e_1,e_2,e_3)\\in \\Gamma(\\partial X,T\\partial X)\\otimes\\mathbb{R}^3$.\nDenote $\\nabla$ the Levi-Civita connection of $g_{\\bm\\omega}$.\n\n\n\n\n\n\\begin{definition}\n$\\bm\\omega$ is a {hypersymplectic} triple if $d\\omega_i=0$. A hypersymplectic triple $\\bm\\omega$ is called {torsion-free} if $d(Q^{ij}\\omega_j)=0$, and is called {hyperk\\\"ahler} if $Q_{ij}=\\delta_{ij}$.\n\\end{definition}\n\nNote that the torsion-free definition coincides with the one defined in the introduction by direct calculation.\n\n\nLet $\\nu$ be the outward unit normal vector field of $\\partial X$. We are going to calculate the second fundamental form of $\\partial X$, say \n$II(v,w)=\\langle\\nabla_v\\nu, w\\rangle_{\\partial X}$.  Let $S\\in\\Gamma(\\partial X,\\text{End}\\ T\\partial X)$ be the shape operator, i.e., $\\langle v,S(w)\\rangle_{\\partial X}=II(v,w)$. Denote $\\Gamma=(\\Gamma_{ij})$ the symmetric matrix $$\\Gamma_{ij}=\\frac{1}{2}\\langle\\gamma_i,d(*_{\\bm\\gamma} \\gamma_j)\\rangle_{\\bm\\gamma}+\\frac{1}{2}\\langle\\gamma_j,d(*_{\\bm\\gamma} \\gamma_i)\\rangle_{\\bm\\gamma}=\\frac{1}{2} *_{\\bm\\eta}(\\eta_i\\wedge d\\eta_j+\\eta_j\\wedge d\\eta_i)  $$ which is completely determined by $\\bm\\gamma$. Denote the matrix $II(e_i,e_j)$ by $A$ and $H=Tr S$.\n\n\n\n\\begin{lemma}\\label{hyperkahler triple second fundamental form}\nIf $\\bm\\omega$ is a hyperk\\\"ahler triple, then \n$$A=\\frac{1}{2}( Tr \\Gamma)I-\\Gamma,$$ In particular, $$2H=*_{\\bm\\eta}(\\bm\\eta\\wedge d\\bm\\eta^T)=\\langle \\gamma_1, d(*_{\\bm\\gamma}\\gamma_1)\\rangle_{\\bm\\gamma}+\\langle \\gamma_2, d(*_{\\bm\\gamma}\\gamma_2)\\rangle_{\\bm\\gamma}+\\langle \\gamma_3, d(*_{\\bm\\gamma}\\gamma_3)\\rangle_{\\bm\\gamma},$$\n$$|S|^2=Tr  {\\Gamma}^2 -H^2.$$\n\\end{lemma}\n\n\\begin{proof}\n\n\n\n Fix a point $p\\in \\partial X$, choose a semi-geodesic coordinate system centered at $p$, say $(x^1,x^2,x^3,t)$, such that a neighborhood of $p$ is identified with $\\{t\\geq 0\\}$\nand its intersection with $\\partial X$ is identified with $\\{t=0\\}$. The hyperk\\\"ahler triple can be written as \n$$\\bm\\omega=-dt\\wedge *_{\\bm\\gamma_t}\\bm\\gamma_t+\\bm\\gamma_t,$$\nwhere $\\bm\\gamma_t$ is a smooth family of closed framings on $\\partial X$ such that $\\bm\\gamma_{0}=\\bm\\gamma$ and $$\\frac{\\partial \\bm\\gamma_t}{\\partial t}=-d(*_{\\bm\\gamma_t}\\bm\\gamma_t).$$\nWe can further choose \n $(x^1,x^2,x^3)$ to be a normal coordinate system for $\\partial X$ at $p$ of the metric $g_{\\bm\\omega}|_{\\partial X}$ such that $e_i=\\partial_{x^i}$ at $p$. Write $\\partial_{x^i}=a_i^k(x,t)e_k(x,t)$, where $\\bm e_k(x,t)=(e_1(x,t),e_2(x,t), e_3(x,t))$ is the dual frame of $\\bm\\eta_t=*_{\\bm\\gamma_t}\\bm\\gamma_t$. Then  at $p$,\n \\begin{equation}\\label{II(ei,ej) calculation}\n      II(e_i,e_j)=II(\\partial_{x^i},\\partial_{x^j})=-\\frac{1}{2}\\partial_t g_{ij}=-\\frac{1}{2}\\partial_t(a_i^ka_j^k)=-\\frac{1}{2}(\\partial_t a_i^k+\\partial_ta_j^k).\n \\end{equation}\nNote that at $p$, $$\\partial_t\\eta_p=\\partial_t a_m^pdx^m=\\partial_t a_m^p\\eta_m,$$ then  we have \n\\begin{equation}\\label{gammaidetaj calculation}\n    \\begin{split}\n        \\langle \\gamma_i,d(*_{\\bm\\gamma}\\gamma_j)\\rangle_{\\bm\\gamma}\n  &=-\\langle\\gamma_i,\\partial_t\\gamma_{j}\\rangle_{\\bm\\gamma}\\\\\n  &=-\\langle \\frac{1}{2}\\delta^{ikl}\\eta_k\\wedge\\eta_l,\\frac{1}{2}\\partial_t(\\delta^{jpq}\\eta_p\\wedge\\eta_q)\\rangle_{\\bm\\gamma}\\\\\n  &=-\\frac{1}{4}\\delta^{ikl}\\delta^{jpq}(\\partial_t a_m^p\\langle \\eta_k\\wedge\\eta_l,\\eta_m\\wedge\\eta_q\\rangle_{\\bm\\gamma}+\\partial_ta_n^q\\langle \\eta_k\\wedge\\eta_l,\\eta_p\\wedge\\eta_n\\rangle_{\\bm\\gamma})\\\\ &=\n-\\frac{1}{4}\\delta^{ikl}\\delta^{jpq}(\\partial_t a_m^p(\\delta^{km}\\delta^{lq}-\\delta^{kq}\\delta^{lm})+\\partial_t a_n^q(\\delta^{kp}\\delta^{ln}-\\delta^{kn}\\delta^{lp}))\\\\&=-\\delta^{ij}\\partial_t a_k^k+\\partial_t a_j^i.\n    \\end{split}\n\\end{equation}\n Combining (\\ref{II(ei,ej) calculation}) and (\\ref{gammaidetaj calculation}), we have \n $$\\Gamma= (Tr A) I-A,$$ so $Tr \\Gamma=2Tr A$, $A=\\frac{1}{2}(Tr \\Gamma) I-\\Gamma$.\n\n\n\\end{proof}\n\n\\section{General Riemannian geometry}\\label{section Riemannian geometry}\n\n\n\\subsection{Evolution equations of hypersurfaces}\nWe refer to \\cite{petersen2016riemannian} Section 3.2 and \\cite{koiso1981hypersurfaces} for discussions in this section.\n\nLet $M$ be a Riemannian manifold, $f$ be a smooth distance function on $M$, i.e., $|\\nabla f|= 1$, so $\\nabla_{\\nabla f}\\nabla f=0$. The $(1,1)$ tensor  corresponds to  $\\text{Hess} f$ is\n\\begin{equation}\\label{(1,1) tensor shape operator}\n    S(X)=\\nabla_X\\nabla f,\n\\end{equation}\n and its trace is $H=\\Delta f\\in C^\\infty(M)$. \n  When $\\Sigma\\subset M$ is a hypersurface defined by a level set of $f$, the restriction of $S$ to $T\\Sigma$ has image in $T\\Sigma$, which is the shape operator of $\\Sigma$. The second fundamental form of $\\Sigma$ with respect to $\\nabla f$ is  $II(X,Y):=\\langle S(X), Y\\rangle=\\text{Hess} f(X,Y)$, so $H$ is the mean curvature and $\\overrightarrow{H}=-H\\nabla f$ is the mean curvature vector.\n  By tensor calculations, we have evolution equations of second fundamental forms\n\n\\begin{equation}\\label{simplified radial curvature equation}\nL_{\\nabla f} S+S^2=-R(\\cdot,\\nabla f)\\nabla f,\n\\end{equation}\n\\begin{equation}\\label{radial curvature equations''}\nL_{\\nabla f} \\text{Hess} f-\\text{Hess}^2 f=-Rm(\\cdot,\\nabla f,\\cdot,\\nabla f),\n\\end{equation}\nwhere $$\\text{Hess}^2 f(X,Y)=\\langle S^2(X),Y \\rangle=\\langle S(X),S(Y)    \\rangle,$$\nand one has the equality\n\\begin{equation}\\label{Lie derivative S same as covariant derivative S}\n    L_{\\nabla f}S=\\nabla_{\\nabla f} S.\n\\end{equation}\nTake the trace of $(\\ref{simplified radial curvature equation})$, we have\n\\begin{equation}\\label{evolution of mean curvature}\nL_{\\nabla f} H=-|S|^2-\\text{Ric}(\\nabla f,\\nabla f). \n\\end{equation}\nwhere $|S|^2:=Tr(S^2)$ is the norm square of the shape operator.\n\nBesides the evolution equations,  we have Gauss equations on $\\Sigma$ \n\\begin{equation}\n    \\begin{split}\n        Rm_{M}(X,W,Y,Z)=&Rm_{\\Sigma}(X,W,Y,Z)+II(X, Z)II(W, Y)\\\\&-II(X, Y)II(W, Z).\n    \\end{split}\n\\end{equation} \nTake the trace with respect to $W,Z$, we have \n\\begin{equation}\\label{trace gauss equation}\n\\text{Ric}_{M}=\\text{Ric}_{\\Sigma}+\\text{Hess}^2 f-H\\cdot \\text{Hess}f+Rm_{M}(\\cdot, \\nabla f,\\cdot,\\nabla f),\n\\end{equation}\ntake the trace again, we have\n\\begin{equation}\\label{scalar curvatures and hypersurface}\n    R_M=R_{\\Sigma}+|S|^2- H^2+2\\text{Ric}_M(\\nabla f,\\nabla f),\n\\end{equation}\n where $R$ denote scalar curvatures.\nUse equations (\\ref{radial curvature equations''}) and (\\ref{trace gauss equation}) to cancel the curvature term involving $\\nabla f$, we get \n\\begin{equation}\\label{evolution equation for 2nd form}\nL_{\\nabla f} \\text{Hess} f=\\text{Ric}_{\\Sigma}-\\text{Ric}_M+ 2\\text{Hess}^2 f-H \\cdot \\text{Hess}f.\n\\end{equation}\n\n\n\\subsection{The boundary exponential map}\n\n\n\nLet $(M,g)$ be a complete Riemannian manifold with boundary, which means the induced metric space is complete. \nDenote $T^\\perp \\partial M$  the normal line bundle of $\\partial M$, which is a trivialized by the inward unit normal vector field $N$. We identify $T^\\perp\\partial M$ with $\\partial M\\times \\mathbb R$ via this trivialization. For $p\\in\\partial M$, denote $\\gamma_p(t)$ the geodesic such that $\\gamma_p(0)=p,\\gamma_p'(0)=N_p$.  Denote $$D(p)=\\inf\\{t>0| \\gamma_p(t)\\in\\partial M\\}\\in (0,\\infty],$$ $$\\tau(p)=\\sup\\{t>0|d(\\gamma_p(t),\\partial M)=t\\}\\in(0,\\infty].$$\nWe have a subset of $U_{\\partial M}\\subset T^\\perp\\partial M$ defined by\n $$U_{\\partial M}=\\{(p,tN_p)\\in T^{\\perp}\\partial M|0\\leq t <D(p) \\},$$ which is the domain of the \\text{boundary exponential map}\n \\begin{equation}\\label{boundary exponential map definition}\n      \\exp^\\perp:U_{\\partial M}\\rightarrow M, (p,s)\\mapsto\\gamma_p(s).\n \\end{equation}\n and define $$V_{\\partial M}=\\{(p,tN_p)\\in T^{\\perp}\\partial M|0\\leq t <\\tau(p) \\}\\subset U_{\\partial M}.$$\n\n\n \n There are some definitions, notations and terminologies related to the boundary exponential map that appear in this paper:\n \\begin{itemize}\n     \\item \n The \\emph{boundary injectivity radius} $i_b$ is defined to be the supremum of $s\\geq 0$ such that $\\exp^\\perp|_{\\partial M\\times[0,s)}$ is a diffeomorphism onto its image.\n \\item A \\emph{focal point} $q$ of $\\partial M$ is a critical value of the boundary exponential map (\\ref{boundary exponential map definition}). If $q$ lies in $\\gamma_p$ for some $p\\in\\partial M$, we say $q$ is a focal point along $\\gamma_p$. \n \\item A \\emph{foot point} of $q\\in M$ is a point  $p\\in \\partial M$ such that $d(q,p)=d(q,\\partial M).$ \n \\item A \\emph{cut point} of $\\partial M$ is a point $q\\in M$ such that there exists a foot point $p$ of $q$ such that $d(q,p)=\\tau(p)$. We also say $q$ is a cut point of $p$.\n \\item When we say a covariant tensor on $M$ is written in \\emph{geodesic gauge}, we mean the pull back of this tensor via $\\exp^\\perp$.\n \\item \n For a  subset $B\\subset\\partial M$, we use the notation  $$C(B,t_1,t_2)=\\exp^\\perp(B\\times[t_1,t_2))$$ to denote a metric cylinder with base $B$.\n \\item $N_r(\\partial M,g\n):=\\{x\\in M|d(x,\\partial M)\\leq r\\}.$\n \\item The $(1,1)$ tensor $S$ in (\\ref{(1,1) tensor shape operator})  is defined with respect to $f=-d(\\cdot,\\partial M)$ near $\\partial M$, so $\\nabla f=-N$ on $\\partial M$, and $II\\geq 0$ if $\\partial M$ is convex.\n \\end{itemize}\n \nHere are some remarks about some of these definitions:\n \n \\begin{itemize}\n     \\item By definition,  $\\gamma_p(s)$ is a focal point of along $\\gamma_p$, if and only if there exists a non-zero \\emph{$\\partial M$-Jacobi field} $V$ along $\\gamma_p$ (a Jacobi field with $ V(0)\\in T_p \\partial M, V'(0)+S(V(0))\\in T_p^\\perp \\partial M$) such that $V(s)=0$. If there is no focal point along $\\gamma_p|_{[0,l)}$, then \n$$I_0(W,W)=\\int_{0}^l \\langle W', W'\\rangle-\\langle R(W,\\gamma_p')\\gamma_p',W \\rangle dt- \\langle S(W),W\\rangle(0)\\geq 0 $$ for any piecewise smooth vector field $W$ along $\\gamma$ with $W(0)\\in T_{\\gamma(0)}\\partial M$. \n\\item \nBy Lemma 3.2 in \\cite{sakurai2017rigidity}, $\\tau$ defines a continuous map from $\\partial M$ to $(0,\\infty]$.  It is well-known that by a second variation argument, the first focal point along $\\gamma_p$ appears no later than $\\tau(p)$, and moreover one can argue by contradiction to get (See Lemma 3.6 in \\cite{sakurai2017rigidity}) \n \\end{itemize}\n \n \n\n\n\\begin{proposition}\\label{first focal point two foot point}\n$q\\in M$ is a cut point of $p$ if and only if at least one of the following holds:\n\\begin{itemize}\n    \\item $q$ is the first focal point of $\\gamma_p$ ;\n    \\item $q$ has at least two foot points.\n\\end{itemize}\n\\end{proposition}\n\\noindent From this,  we have $\\exp^\\perp|_{V_{\\partial M}}$ is a diffeomorphism and\n$$i_b=\\inf_{p\\in\\partial M}\\tau (p).$$\n\n\nIt is worth noting that if $M$ is embedded in some complete Riemannian manifold $M'$ of the same dimension and view $\\partial M$ as an embedded hypersurface of $M'$, then it may happen that a ``focal point'' of $\\partial M$ in $M'$ lies outside $M$ if we define the boundary exponential map in the whole normal bundle $T^\\perp\\partial M$. However, our definition is intrinsic for $M$.\n\n\n\n\\subsection{Manifolds with mean convex boundary}\n\nNow we focus on manifolds with mean convex boundary. We will summarize some results and discuss a new result.\n\nThe following results are well-known,  see for example \\cite{li2014sharp} and \\cite{Donaldson2017BoundaryVP}.\n\n\n\\begin{proposition}\\label{diam}\nLet $(M,g)$ be a compact, connected Riemannian manifold with boundary, $\\emph{Ric}_M\\geq 0$. Suppose $\\partial M$ has mean curvature $H\\geq H_0>0$, then $\\partial M$ is connected,\n\n$$\\pi_1(M,\\partial M)=0,$$ $$\\sup_{q\\in M}d(q,\\partial M)\\leq (n-1)H_0^{-1},$$  $$\\emph{vol}(M)\\leq C(n)H_0^{-1}\\emph{vol}(\\partial M).$$\n\\end{proposition}\n\n\n\n\n\n\n\n\\begin{proof}\n\nIf $\\pi_1(M,\\partial M)\\neq 0$ or $\\pi_0(\\partial M)\\neq 0$, then every non-trivial class contains a non-trivial unit speed geodesic $\\gamma:[0,l]\\rightarrow M $ which minimize the length of all curves in its class.\nFrom the first variation formula, $\\gamma$ intersects boundary perpendicularly at both end points. Pick an orthonormal basis $V_i,1\\leq i\\leq n-1$ of $T_{\\gamma(0)}\\partial M$ and parallel them transport along $\\gamma$ to get $V_i(t)$. Let $\\gamma_{i,s}(t)$ be a family of curves centered at $\\gamma $ with variation field $V_i(t)$. By second variational formula, $$0\\leq \\sum_{i=1}^{n-1} \\frac{d^2}{ds^2}\\Big{|}_{s=0}E(\\gamma_{i,s})=\\int_0^l -\\text{Ric}(V_i(t),V_i(t))dt-H(\\gamma(0))-H(\\gamma(l))<0,$$ which is a contradiction.\n\nIf for some $q\\in M$, $\\tilde{l}:=d(q,\\partial M)> (n-1)H_0^{-1}$. Let $p$ be a foot point of $q$. Let $V_i,1\\leq i\\leq n-1$ be an orthonormal basis of $T_{p}\\partial M$ and parallel them transport along $\\gamma_p$ to get $V_i(t)$, and denote $\\tilde{V}_i(t)=(\\tilde l-t)V_i(t)$, $\\tilde \\gamma_{i,s}(t)$  a family of curves centered at $\\gamma_p $ with variation field $\\tilde{V}_i(t)$, then $$0\\leq \\sum\\limits_{i=1}^{n-1}\\frac{d^2}{ds^2}\\Big{|}_{s=0}E(\\tilde\\gamma_{i,s})=(n-1)\\tilde l-\\int_0^{\\tilde{l}} \\text{Ric}(tV_i(t),tV_i(t))dt-\\tilde{l}^2H(\\gamma(0))<0,$$ which is a contradiction.\nThe volume upper bound is by volume comparison, see \\cite{heintze1978general}.\n\\end{proof}\n\n\\begin{remark}\nNote that when $M$ is connected,   $\\pi_1({M},\\partial M)=0$ is equivalent to that $\\partial M$ is connected and the natural map $\\pi_1(\\partial M)\\rightarrow \\pi_1(M)$ is surjective, the latter of which means that fix a point $p_0\\in\\partial M$, then\nfor any closed path $x(t):0\\leq t\\leq 1$  in $M$ with $x(0)=x(1)=p_0$, there exists a homotopy $F_s(t):0\\leq s, t\\leq 1$ with $F_s(0)=F_s(1)=p_0, F_0(t)=x(t)$ such that $F_1(t)\\in \\partial M$.\n\\end{remark}\n\n\n\n\n\n\nThe following result is well-known, see Lemma 6.3 of \\cite{kodani1990convergence}.\n\n\n\\begin{proposition}\nLet $M$ be a complete Riemannian manifold with  nonempty  compact boundary. If there are no focal point whose distance to $\\partial M$ is equal to $i_b$, then there exists a smooth geodesic of length $2i_b$ which is perpendicular to $\\partial M$ at both end points.\n\\end{proposition}\n\n\\begin{proof}\n$i_b=\\inf\\limits_{p\\in\\partial M}\\tau (p)>0$. Suppose the infimum is achieved at $p_1\\in\\partial M$. Then by assumption and Proposition \\ref{first focal point two foot point} $\\gamma_{p_1}(i_b)$ has another foot point $p_2$. We claim $\\gamma_{p_1}'(i_b)=-\\gamma_{p_2}'(i_b)$, so $D(p_1)=D(p_2)=2i_b$ and $\\gamma_{p_1}:[0,2i_b]\\rightarrow M$ is the smooth geodesic we want. By assumption, we can find smooth distance functions $h_1, h_2$  extending $d(\\cdot,\\partial M)$  near $\\gamma_{p_1}|_{[0,i_b]}$, $\\gamma_{p_2}|_{[0,i_b]}$, respectively. Consider the smooth hypersurface $\\Sigma=(h_1-h_2)^{-1}(0)$ near $q:=\\gamma_{p_1}(i_b)=\\gamma_{p_2}(i_b)$. Then $v=\\nabla h_1(q)+\\nabla h_2(q)\\in T_{q}\\Sigma$. If it is non-zero, then $\\langle \\nabla h_1(q)+\\nabla h_2(q) , v \\rangle>0$. Without loss of generality, assume $\\langle\\nabla h_1,v\\rangle>0$. Then in the direction of $-v$ in $\\Sigma$, we have some point $q'\\in\\Sigma$ with $h_1(q')<h_1(q)$. Hence $q'$ has two foot points and $d(q',\\partial M)=h_1(q')<i_b$, which is a contradiction.\n\\end{proof}\nIt is worth noting that in this proof, Kodani used the first order variation of $h_1$ on $\\Sigma$ to lead a contradiction. We can also investigate the second order variation of $h_1$ on $\\Sigma$ and prove the following:\n\n\n\n\n\n\n\n\n\n\\begin{proposition}\\label{existence of focal points}\nLet $M$ be a compact Riemannian manifold with mean convex boundary, $\\emph{Ric}_M\\geq 0$,  then there exists a focal point of $\\partial M$ whose distance to $\\partial M$ is equal to $i_b$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose not, by the previous proposition, we have a smooth geodesic of length $2i_b$ which is perpendicular to $\\partial M$ at both end points. We use notations $p_1,p_2, q, h_1,h_2,\\Sigma$ as in the previous proposition. We claim $\\Delta_\\Sigma h_1(q)<0$, so we get another point $q''\\in \\Sigma$ near $p$ with $h_1(q'')<h_1(q)=i_b$ and get a contradiction. Denote $N_{0}=\\nabla h_1(q)=-\\nabla h_2(q),\\Sigma_1=h_1^{-1}(i_b),\\Sigma_2=h_2^{-1}(i_b)$, then $N_0$ is a common unit normal vector for $\\Sigma,\\Sigma_1,\\Sigma_2$ at $q$. Let $II_\\Sigma, II_{\\Sigma_1},II_{\\Sigma_2}$ be second fundamental forms with respect to $N_0$ at $q$.\nThen at $q$, $$II_\\Sigma= \\frac{1}{|\\nabla (h_1-h_2)|}\\text{Hess} (h_1-h_2)=\\frac{1}{2} \\text{Hess}(h_1-h_2)=\\frac{1}{2}(II_{\\Sigma_1}+II_{\\Sigma_2}),$$  hence \n $$H_\\Sigma=\\frac{1}{2}(H_{\\Sigma_1}+H_{\\Sigma_2}),\n\\overrightarrow{H}_{\\Sigma}=\\frac{1}{2}(\\overrightarrow{H}_{\\Sigma_1}+\\overrightarrow{H}_{\\Sigma_2}).$$From the formula of Laplace operator on a hypersurface, we know that at $q$,\n$$\\Delta_\\Sigma h_1=\\Delta h_1-\\text{Hess} h_1(N_0,N_0)+\\langle \\nabla h_1, \\overrightarrow{H}_\\Sigma \\rangle ,$$\n$$\\Delta_{\\Sigma_1} h_1=\\Delta h_1-\\text{Hess} h_1(N_0,N_0)+\\langle \\nabla h_1,\\overrightarrow{H}_{\\Sigma_1} \\rangle.$$\nSince $h_1$ is a constant on $\\Sigma_1$, $\\Delta_{\\Sigma_1}h_1=0$, hence\n\\begin{equation}\n\\begin{split}\n\\Delta_{\\Sigma} h_1 &=\\Delta_{\\Sigma} h_1-\\Delta_{\\Sigma_1}h_1=\\langle\\nabla h_1, \\overrightarrow H_{\\Sigma}-\\overrightarrow H_{\\Sigma_1}\\rangle=\\langle \\nabla h_1, \\frac{1}{2}(\\overrightarrow H_{\\Sigma_2}-\\overrightarrow H_{\\Sigma_1})\\rangle \\\\\n&=-\\frac{1}{2}(H_{\\Sigma_2}-H_{\\Sigma_1}).\n\\end{split}\n\\end{equation}\nSince $\\partial M$ is mean convex, $\\text{Ric}_M\\geq 0$, by the evolution equation of mean curvature (\\ref{evolution of mean curvature}), we have $-H_{\\Sigma_1}(q)> H_{\\partial M}(p_1)>0$, $H_{\\Sigma_2}(q)> H_{\\partial M}(p_2)>0$. Hence $\\Delta_{\\Sigma} h_1(p)<0$, which completes the proof.\n\n\n\\end{proof}\n\nA moment thought about the arguments in the end of the previous proof yields that $\\text{Ric}_M\\geq 0$ is not so necessary, since we can make use of the evolution equation (\\ref{evolution of mean curvature}) to get an ordinary differential inequality for the mean curvature. \n\n\\begin{proposition}\\label{more general inj lower bound}\nIf in the previous proposition we assume instead $\\emph{Ric}_M\\geq -(n-1)c$ for some $c>0$, and $H\\geq  H_0>0$. If $i_b< -\\frac{1}{2}(n-1)\\ln \\big{|}\\frac{H_0-(n-1)\\sqrt{c}}{H_0+(n-1)\\sqrt{c}}\\big{|}$, then \nthere exists a focal point of $\\partial M$ whose distance to $\\partial M$ is equal to $i_b$. \n\\end{proposition}\n\n\\begin{proof} Suppose the conclusion is not true, follow the arguments as before except for second last sentence. Let  $S_i(X)=-\\nabla_{X}\\nabla h_i$, $H_i=Tr S_i=-\\Delta h_i$, and identify a neighborhood of $\\gamma_{p_i}|_{[0,i_b]}$ with a subset of $\\partial M\\times\\mathbb{R}$ via $\\exp^\\perp$. Then by (\\ref{evolution of mean curvature}),$$\\partial_t{ H_i}=|S_i|^2+\\text{Ric}(\\nabla h_i,\\nabla h_i)\\geq\\frac{1}{n-1}H_i^2-(n-1)c$$ and $H(p_i,0)\\geq H_0$.\nLet $f$ solves the ODE on $[0,i_b]$\n $$ f'=\\frac{1}{n-1}f^2-(n-1)c$$ and $f(0)=H_0$, then we have $H_i(p_i,t)\\geq f(t)  $.  In particular, $H_i(p_i,i_b)\\geq f(i_b)>0$, which leads to a contradiction as before.\n\n\\end{proof}\n\n\n\nProposition $\\ref{existence of focal points}$ implies\n\n\\begin{corollary}\\label{Ric positive, mean positive, inj lower bound}\nLet $(M,g)$ be a compact Riemannian manifold with boundary, $K>0,\\lambda>0$ are constants.\nSuppose $\\sec \\leq K$, $S\\leq \\lambda$, $H>0$, $\\emph{Ric}_M \\geq 0$, then $i_b\\geq \\frac{1}{\\sqrt{K}}\\emph{arccot}{\\frac{\\lambda}{\\sqrt{K}}}.  $\n\\end{corollary}\n\n\n\\begin{proof}\nBy Proposition \\ref{existence of focal points}, there exists $p\\in\\partial M$ such that $\\gamma_p(i_b)$ is a focal point along $\\gamma_p$. If $i_b<\\frac{1}{\\sqrt{K}}\\text{arccot}{\\frac{\\lambda}{\\sqrt{K}}} $, from comparison theorem for Jacobi fields, we know  $\\gamma_p(i_b)$ cannot be a focal point along $\\gamma_p$, which is a contradiction. \n\\end{proof}\n\nSimilarly, Proposition \\ref{more general inj lower bound} implies\n\n\\begin{corollary}\\label{inj lower bound limit}\nLet $(M,g)$ be a compact Riemannian manifold with boundary. Suppose $|Rm|\\leq C$, $|S|\\leq C$, $H\\geq H_0>0$, then we can find $i_0$ depending explicitly on $C,H_0$ such that $i_b\\geq i_0.$\n\\end{corollary}\n\n\\begin{remark}\\label{inj lower bound limit remark}\nIn the previous two corollaries, if the sectional curvature and Ricci curvature bounds only holds for $N_1(\\partial M,g)$, then we also have a $i_b$ lower bound. In the case of Corollary \\ref{Ric positive, mean positive, inj lower bound}, we have $i_b\\geq \\min\\{\\frac{1}{\\sqrt{K}}\\text{arccot}{\\frac{\\lambda}{\\sqrt{K}}} ,1\\}$. \n\\end{remark}\n\n\n\n\n\\begin{remark}\nIn the same setting as the previous two corollaries, \\cite{knox2012compactness} Lemma 2.2 claimed to prove a lower bound for $i_b$, using a similar method as \\cite{Anderson2012BoundaryVP} Lemma 2.4.  In both papers, there is a logic problem that they get a contradiction with an unjustified statement: Let $M$ be a Riemannian manifold with boundary, $\\gamma:[0,l]\\rightarrow M$ be a geodesic that is perpendicular to the boundary at both end points, and suppose there is no focal point along $\\gamma$ for both boundary portions, then $I_1(V,V)\\geq 0$ for any smooth vector field along $\\gamma$ with $V(0),V(l)\\in T{\\partial M}$. Here $$I_1(V,V)=\\int_{0}^l \\langle V', V'\\rangle-\\langle R(V,\\gamma')\\gamma',V \\rangle dt- \\langle S(V(0)),V(0)\\rangle-\\langle S(V(l)),V(l)\\rangle. $$ In fact, this unjustified statement is not true, and one can easily think of an example: let $$\\Sigma_1=\\{(x',x^n)\\in\\mathbb{R}^n\\big | |x'|^2+(1-x^n)^2=R_1^2\\},$$ $$\\Sigma_2=\\{(x',x^n)\\in\\mathbb{R}^n\\big | |x'|^2+(1+x^n)^2=R_2^2\\},$$ $R_1,R_2>2$\nand $\\gamma(t)=(0,1-t), 0\\leq t\\leq 2 $, then $I_1(V,V)=-\\frac{1}{R_1}-\\frac{1}{R_2\n}<0$ for any unit-norm parallel vector field along $\\gamma$ with $V(0)\\in T_{\\gamma(0)}\\Sigma_1$. In this case, there exist no focal points on $\\gamma$ for both $\\Sigma_1,\\Sigma_2$.\n\nIn fact, focal points give crucial information for index form defined by one submanifold and one point. However, as seen from the example, they do not fit well with the index form defined for two submanifolds. Indeed, there is some notion of ``conjugate point'' defined for two submanifolds, see \\cite{ambrose1961index}. \n\\end{remark}\n\n\n\n\n\nIt is easy to see focal points can ``pass to the limit'', since they arise from kernels of the differential of exponential maps. Though one can use Corollary \\ref{inj lower bound limit} directly in many situations, we point out this fact here, which may help in contradiction arguments.\n\n\\begin{proposition}\\label{pass to limit}\nLet $M$ be a manifold with boundary, $g_i$ be a sequence of Riemannian metrics on $M$, and $p_i\\in\\partial M$. Suppose $g_i$ converges to a Riemannian metric $g_{\\infty}$ smoothly and $p_i$ converges to $p_\\infty\\in\\partial M$, $\\gamma_{p_i}$ is defined on $[0,b]$ and\n$\\gamma_{p_i}(t_i)$ is a focal point along $\\gamma_{p_i}$ with $0<a\\leq t_i\\leq b$. Then for a subsequence, $\\gamma_{p_i}$ converges smoothly to $\\gamma_{p_\\infty}$, $t_i\\rightarrow t_\\infty$ and $\\gamma_{p_\\infty}(t_\\infty)$ is a focal point along $\\gamma_{p_\\infty}$.\n\\end{proposition}\n\n\\begin{proof}\n$\\gamma_{p_i}'(0)\\in TM$ is a bounded sequence, hence subconverges to some $v\\in TM $, which must equal to $\\gamma_{p_\\infty}'(0)$. Hence by ODE theories, $\\gamma_{p_i}$ converges smoothly to $\\gamma_{p_\\infty}$. Suppose for a subsequence $t_i\\rightarrow t_\\infty$. Let $J_i:[0,t_i]\\rightarrow TM$ be $\\partial M$-Jacobi fields along $\\gamma_i$ with $J_i(t_i)=0, |J_i'(t_i)|=\\frac{t_\\infty}{t_i}$. Normalize these geodesics and Jacobi-fields by $\\bar\\gamma_i(t)={\\gamma}_i(\\frac{t_i}{t_\\infty} t)$, $\\bar{J}_i(t)=J_i(\\frac{t_i}{t_\\infty} t)$, so  $\\bar\\gamma_i, \\bar{J}_i$ are defined on the same interval $[0,t_\\infty]$, and\n\\begin{equation}\n\\begin{split}\n\\bar J_i''(t)&+  Rm_{g_i}(\\bar{J}_i(t),\\bar\\gamma_i'(t))\\bar\\gamma_i'(t)=0,\\\\\n \\bar{J}_i(t_\\infty)&=0,|\\bar{J}_i'(t_\\infty)|=1.\n\\end{split}\\end{equation}\nFor a subsequence $\\bar{J_i}'(t_\\infty)\\rightarrow w$ with $|w|=1$. Let ${J_\\infty}$ be the non-trivial Jacobi-field along $\\gamma_{p_\\infty}$ with $J_\\infty(t_\\infty)=0, J'_\\infty(t_\\infty)=w$, then $J_i$ converges smoothly as maps $[0,t_\\infty]\\rightarrow TM$ to $J_\\infty$ by ODE theories. Hence $J_\\infty$ is a $\\partial M$-Jacobi field, which implies $\\gamma_{p_\\infty}(t_\\infty)$ is a focal point along $\\gamma_{p_\\infty}.$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\\subsection{Volume estimates near boundary}\n\nIn this subsection, we show some volume lower bounds near boundary under some geometric control. These estimates can be found in \\cite{anderson2004boundary} and \\cite{knox2012compactness}.\n\n\n\\begin{proposition}\\label{volume of a metric cyclinder}\nLet $({M},g)$ is a Riemannian manifold with boundary, suppose $\\emph{Ric}\\geq -(n-1)c$ for some $c\\geq 0$, and $\\exp^\\perp$ is an diffeomorphism in $B\\times [0,T)$ for some open subset $B$ of $\\partial M$, then $$\\sup_{B\\times \\{t\\}} H\\leq\\max\\{(n-1)\\sqrt{2c},4(n-1)^{-1}T^{-1}\\},$$\n$$\\emph{vol}(C(B,t_1,t_2))\\geq C(n,T,c)\\emph{vol}_{\\partial M}(B)|t_2-t_1|,$$ where $0\\leq t, t_1,t_2\\leq\\frac{1}{2}T.$\n\n\\end{proposition}\n\n\\begin{proof}\nBy the evolution equation (\\ref{evolution of mean curvature}),\n\\begin{equation}\n\\partial_t H=|S|^2+\\text{Ric}(N_t,N_t),\n\\end{equation}\nhence\n\\begin{equation}\n\\partial_t H\\geq \\frac{1}{n-1}H^2-(n-1)c.\n\\end{equation}\nIf for some $t_0\\in[0,\\frac{1}{2}T],z_0\\in B$, we have $H(z_0,t_0)\\geq \\delta$ and $\\delta\\geq (n-1)\\sqrt{2c}$, then $\\partial_t H(z_0,t)\\geq 0$ and $H(z_0,t)\\geq (n-1)\\sqrt{2c}$ for $t\\in[t_0,T) $. Hence $$\\partial_t H(z_0,t)\\geq \\frac{1}{2}(n-1)^{-1}H(z_0,t)^2,$$ $$H^{-1}(z_0,t)\\leq H^{-1}(z_0,t_0)-\\frac{1}{2}(n-1)(t-t_0)\\leq \\delta^{-1}-\\frac{1}{2}(n-1)(t-\\frac{1}{2}T).$$ Then the continuity of $H(z_0,\\cdot)$  in  $[0,T)$ forces $\\delta\\leq 4(n-1)^{-1}T^{-1}$, which proves the mean curvature estimate.\n\n\nFor $t\\in[0,\\frac{1}{2}T]$,  let $B_t=\\exp^\\perp(B\\times \\{t\\})$, then $$\\frac{d}{dt}\\mathcal{H}^{n-1}(B_t)=-\\int_{B_t} Hd\\mathcal{H}^{n-1}(B_t)\\geq-C_1(n,T,c)\\mathcal{H}^{n-1}(B_t) ,$$where $\\mathcal{H}^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure of ${M}$. Hence $$\\mathcal{H}^{n-1}(B_t)\\geq e^{-C_1(n,T,c)t}\\text{vol}_{\\partial M}(B)\\geq e^{-\\frac{1}{2}C_1(n,T,c)T}\\text{vol}_{\\partial M}(B)$$ and when $0\\leq t_1<t_2\\leq \\frac{1}{2}T$, $$\\text{vol}(C(B,t_1,t_2))=\\int_{t_1}^{t_2}\\mathcal{H}^{n-1}(B_t)dt\\geq C(n,T,c)\\text{vol}_{\\partial M}(B)(t_2-t_1).$$\n\\end{proof}\n\n\n\n\n\n\\begin{proposition}\\label{volume noncollasping}\nLet $M$ be a Riemannian manifold with boundary, $p\\in\\partial M$. Suppose  $B(p,2r_0)$ has compact closure,\n$$\\sup\\limits_{B(p,2r_0)}|Rm|\\leq C,$$\n$$\\emph{vol}_{\\partial M}(B_{\\partial M}(p,r_0))\\geq v_0,$$\n  $\\exp^\\perp$ is an diffeomorphism in $B_{\\partial M}(p,r_0)\\times [0,r_0)$, and on $B_{\\partial M}(p,r_0)$ $$  \\emph{Ric}_{\\partial M}\\geq -(n-2)c_0,  \\ |S|\\leq C,$$  then there exists $r_1>0,v_1>0$ depending on $n, C,c_0, r_0,v_0$, such that for $q=\\exp^\\perp(p,2r_1)$, we have $$\\emph{vol}(B(q,r_1))\\geq v_1.$$\n\n\\end{proposition}\n\n\\begin{proof}\n\n By definition $C(B_{\\partial M}(p,r_0),0,r_0)\\subset B(p,2r_0)$.  We claim that  there exists $r_2>0,C_1>0$ such that for any $p_1,p_2\\in B_{\\partial M}(p,r_0)$, $$d_{\\Sigma_t}(\\exp^\\perp(p_1,t),\\exp^\\perp(p_2,t))\\leq C_1d_{\\partial M}(p_1,p_2)$$ when $0\\leq t\\leq r_2$, where $\\Sigma_t$ is the image of $B_{\\partial M}(p,2r_0)$ under $\\exp^\\perp(\\cdot, t)$. In fact, by (\\ref{simplified radial curvature equation})(\\ref{Lie derivative S same as covariant derivative S}),\n $$\\nabla_{\\nabla t } S=S^2+R(\\cdot,\\nabla t)\\nabla t, $$ hence  $$\\frac{d}{dt}|S|\\leq |S|^2+C.$$\nIntegrate the inequality, we have \n$$\\arctan(\\frac{|S|}{\\sqrt C})(x,t)-\\arctan(\\frac{|S|}{\\sqrt C})(x,0)\\leq \\sqrt{C}t.$$\nHence there exist $r_2>0,C_2>1$ such that $|S|\\leq \\log C_2$ when $0\\leq t\\leq r_2$.\nNow let $\\gamma_0(s)$ be a smooth curve in $B_{\\partial M}(p,2r_0)$ that connects $p_1,p_2$, and let $\\gamma_t(s)=\\exp^\\perp(\\gamma_0(s),t)\\in \\Sigma_t$, then we have $$\\frac{d}{dt} \\log |\\gamma_t'(s)|=-\\frac{\\langle S_{\\Sigma_t}(\\gamma_t'(s)),\\gamma_t'(s)\\rangle}{\\langle\\gamma_t'(s),\\gamma_t'(s)\\rangle}\\leq |S_{\\Sigma_t}(\\gamma_t(s))|\\leq \\log C_2.$$\nIt follows that $|\\gamma_t'(s)|\\leq C_1|\\gamma_0'(s)|$ with $C_1=C_2^{r_2}$ and the claim follows from integration. \nNow take $$r_1=\\min\\{2C_1r_0,\\frac{1}{4}r_2\\},$$ we have $$C(B_{\\partial M}(p,\\frac{r_1}{2C_1}),\\frac{3r_1}{2},\\frac{5r_1}{2})\\subset B(q,r_1),$$ then we apply Proposition \\ref{volume of a metric cyclinder} and Bishop-Gromov volume comparison on $\\partial M$ to get the desired conclusion.\n\\end{proof}\n\n\\begin{remark}\nIt is easy to give a quantitative version of the lemma from the proof. However, to the author's knowledge, we cannot prove the last inclusion in the proof without a control of curvature. It may be possible that a metric ball of the boundary becomes ``long and thin'' under the flow of $\\nabla d(\\cdot,\\partial M)$,  while maintains an area lower bound. \n\\end{remark}\n\n\n\n\\subsection{Harmonic radius and convergence theory}\n\nConvergence theory of Riemannian manifolds is a powerful tool to prove conclusions in Riemannian geometry through contradiction arguments when explicit bounds is not required. In this section, we will restate some results of \\cite{anderson2004boundary}, follow the proof there, and discuss some direct corollaries.\n\nLet $(M,g)$ be a Riemannian manifold with boundary, $m\\in \\mathbb{N}$,   $0<\\alpha<1$, $Q>1$. For $p\\in M$, define $r_h^{m,\\alpha}(p,g,Q)$ to be the supremum of $\\rho>0$ such that if $d(p,\\partial M)>\\rho$,\nthen there exists a neighborhood $U$ of $p$ in $M$ and a interior coordinate chart $\\varphi:B_{\\frac{\\rho}{2}}\\rightarrow U$, $\\varphi(0)=p$, and  if $d(p,\\partial M)\\leq\\rho$, then there exists a neighborhood $U$ of $p$ in $M$ and a boundary coordinate chart\n$\\varphi: B_{4\\rho}^+\\rightarrow U$, $\\varphi((0,d(p,\\partial M)))=p, \\varphi(\\tilde{B}_{4\\rho})=U\\cap\\partial M$, and in either $B_{\\frac{\\rho}{2}}$ or $B_{4\\rho}^+$, we have $$\\Delta_M\\varphi^{-1}=0,$$\n$$Q^{-2}(\\delta_{ij})\\leq (g_{ij})\\leq Q^2 (\\delta_{ij}), $$\n$$\\rho^{m+\\alpha}\\sum_{|\\beta|=m}|\\partial_\\beta g_{ij}(x)-\\partial_\\beta g_{ij}(y)|\\leq (Q-1)|x-y|^\\alpha$$\nWe call such a coordinate chart a $(\\rho,Q,m,\\alpha)$-harmonic coordinate chart centered at $p$. Note that the second condition implies there exists $r_1,r_2$, depending on $\\rho,Q$, such that\n$B(p,r_1)\\subset U\\subset B(p,r_2)$.\n\n\n\n\n\\begin{definition}\nFix an integer $m\\geq 0$, and $0<\\alpha<1$.  We say a sequence of Riemannian manifold with boundary $(M_i,g_i,p_i)$ converges in pointed $C^{m,\\alpha}$ to $(M_\\infty,g_\\infty,p_\\infty)$ if there exists precompact open subsets  $\\Omega_{i}$ of $M_i$ and $\\Omega_{\\infty,i}$ of $M_\\infty$, and $\\sigma_i>\\rho_i\\rightarrow\\infty$ such that $B(p_i,\\rho_i)\\subset \\bar{\\Omega}_{i}\\subset B(p_i,\\sigma_i)$, $B(p_\\infty,\\rho_i)\\subset \\bar{\\Omega}_{i}\\subset B(p_\\infty,\\sigma_i)$ and  there exists diffeomorphisms $F_{i}:\\Omega_{\\infty,i}\\rightarrow \\Omega_{i}$, $F_{i}:\\Omega_{\\infty,i}\\cap\\partial M_i\\rightarrow\\Omega_{i}\\cap\\partial M_\\infty$ such that  $F_{i}^*g_i\\rightarrow g$ in $C^{m,\\alpha}$ topology,  and $F_{i}^{-1}(p_i)\\rightarrow p_\\infty$. If we replace $C^{m,\\alpha}$ by $C^\\infty$, we say the convergence is in pointed Cheeger-Gromov sense.\n\\end{definition}\n\n\\begin{remark}\n$(M_\\infty, g_\\infty)$ is automatically a complete $C^{m,\\alpha}$ or $C^\\infty$ Riemannian manifold with boundary from the definitions. Sometimes we only need that one metric ball converges, so one can modify the definitions above: suppose $\\bar{B}(p_i,r)\\subset \\Omega_i$ for some precompact open set $\\Omega_i\\subset M_i$ and there exists a Riemannian manifold with boundary $(\\Omega_\\infty,g_\\infty)$, a point $p_\\infty\\in\\Omega_\\infty$, and  diffeomorphisms $F_i:\\Omega_{\\infty}\\rightarrow\\Omega_i$ mapping $\\partial \\Omega_\\infty$ onto $\\Omega_i\\cap\\partial M_i$ such that $F_i^*g_i\\rightarrow g_\\infty$ in $C^{m,\\alpha} $ or $C^\\infty$ topology and $F_i^{-1}(p_\\infty)\\rightarrow p_i$, we say $B(p_i,r)$ converges in $C^{m,\\alpha}$  or Cheeger-Gromov sense to $B(p_\\infty, r)$.\n\n\\end{remark}\n\n\n\n\n\n\n\n\n\nThe following theorem is well-known and is a fundamental theorem of Riemannian convergence theory. \n\n\\begin{proposition}\\label{harmonic radius lower bound}\nLet $(M_i,g_i)$ be  a sequence of  complete Riemannian manifold with boundary, $p_i\\in M_i$. Suppose there exists some $Q>1$, and a positive function $r:(0,\\infty)\\rightarrow(0,\\infty) $, such that $r_h^{m,\\alpha}(p,g_i,Q)\\geq r(R)$ for any $p\\in B(p_i,R)$, then for a subsequence, $(M_i,g_i,p_i)$ converges in pointed $C^{m,\\beta}$ sense to $(M_\\infty,g_\\infty,p_\\infty)$ for any $0<\\beta<\\alpha$. If the above assumption holds for only one $R$, then $B(p_i,R)$ converges in $C^{m,\\beta}$ sense to $B(p_\\infty, R)$.\n\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\nNext, we discuss under what geometric control we can  get a harmonic radius lower bound. We state and prove the following local version of Theorem 3.2.1 in  \\cite{anderson2004boundary}, with simplified arguments in some parts.\n\n\n\n\\begin{theorem}\\label{local harmonic radius lower bound}\nFix $m\\geq 1$. Let $(M,g)$ be a  Riemannian manifold with boundary, and $\\Sigma\\subset\\partial M$ be a boundary metric ball with compact closure, nonempty boundary. Suppose $\\exp^\\perp$ maps $\\Sigma\\times [0,i_0)$ diffeomorphically onto its image $\\Omega$,  \n\\begin{equation}\\label{injectivity radius explaination}\n     inj_{\\Omega}\\geq i_0,\\ inj_{\\Sigma}\\geq i_0,\n\\end{equation}\nin $\\Omega$,\n\\begin{equation}\\label{bounds 1 up to m}\n    |\\nabla^l \\text{Ric}_M|\\leq \\Lambda, 0\\leq l\\leq m,\n\\end{equation}\n on $\\Sigma$\n \\begin{equation}\\label{bounds 2 up to m}\n     |\\nabla_{\\partial M}^l \\text{Ric}_{\\partial M}|\\leq \\Lambda,   |\\nabla_{\\partial M}^{l+1} H|\\leq \\Lambda, 0\\leq l\\leq m.\n \\end{equation}\n Then for any $Q>1$, $\\alpha\\in(0,1)$, $p\\in\\Omega$, \n \\begin{equation}\\label{harmonic radius lower bound non empty boundary}\n     r^{m+1,\\alpha}_h(p,g,Q)\\geq r_0(i_0,\\Lambda,m,\\alpha,Q)d(p,\\partial ^+\\Omega),\n \\end{equation}\n  where $\\partial^+\\Omega=\\bar {\\Omega} \\backslash(\\Omega\\cup\\partial M)$.\n\\end{theorem}\n\n\n\n\n\\begin{remark}\\label{local harmonic radius lower bound remark} One should understand (\\ref{injectivity radius explaination}) as follows:\nfor an open set $U$ inside a Riemannian manifold with boundary, $inj_U\\geq i_0$ means for each $p\\in U$, $\\exp_{p}$ maps $B_{i_0}(0)\\subset T_p M$ diffeomorphically onto its image if $d(p,U^c)\\geq i_0$, and maps $B_{\\frac{1}{2}d(p,U^c)}(0)\\subset T_p M$ diffeomorphically onto its image if $d(p,U^c)\\leq i_0$.\n\\end{remark}\n\n\n\n\\begin{proof}\nIf not, we have a sequence $(M_k,\\tilde{g}_k)$ and $\\Sigma_k$, $\\Omega_k$ that satisfies the conditions, but there exists $p_k\\in \\Omega_k$ with\n$$\\frac{r_h^{m+1,\\alpha}(p_k,\\tilde{g}_k,Q)}{d_{\\tilde{g}_k}(p_k,\\partial^+ \\Omega_k)}=\\inf_{p\\in \\Omega_k} \\frac{r_h^{m+1,\\alpha}(p_k,\\tilde{g}_k,Q)}{d_{\\tilde{g}_k}(p,\\partial^+ \\Omega_k)}\\rightarrow 0.$$  Rescale the metric $g_k=(r_h^{m+1,\\alpha}(p_k,\\tilde g_k,Q))^{-2}\\tilde{g}_k$, so $r_h^{m+1,\\alpha}(p_k,{g}_k,Q)=1$, then $d_{g_k}(p_k,\\partial^+\\Omega_k)\\rightarrow\\infty$, and $r_h^{m+1,\\alpha}(p,{g}_k,Q)\\geq\\frac{1}{2}$ if $d_{g_k}(p,p_k)\\leq R$ , $k\\geq k(R)$. Fix any $\\beta\\in(0,\\alpha)$. Then there are two cases: \n \n\\textbf{Case 1}\n $d_{g_k}(p_k,\\Sigma_k)\\rightarrow\\infty$ for some subsequence.\n \n \nThen a subsequence $(M_k,{g_k},p_k)$ converges in pointed $C^{m+1,\\beta}$ sense to a complete Riemannian manifold $(M_\\infty,g_\\infty,p_\\infty)$.  So $\\text{Ric}_{M_\\infty}=0$, $inj_{M_\\infty}=\\infty$. By Cheeger-Gromoll splitting theorem, $(M_\\infty, g_\\infty)$ is isometric to $(\\mathbb{R}^n, g_{flat})$. \n\nHence for any $L>0$, there exist a coordinate $\\varphi_{0,k}:B_{L+5} \\rightarrow U_k\\subset M_k$, $\\varphi_{0,k}(0)=p_k$,  such that\n$$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\beta}(B_{L+5})}\\rightarrow 0, 1\\leq i,j\\leq n.$$ \nWe solve the Dirichlet problem for functions $u_k^\\nu$, $1\\leq\\nu\\leq n$:\n\n\\begin{equation}\n\\Delta_{M_k}u_k^\\nu=0 \\ {\\rm in} \\ {B}_{L+5} , u_k^\\nu|_{\\partial {B}_{L+5}}=x^\\nu\\ .\n\\end{equation}\nRecall the formula $$\\Delta_{g}=g^{ij}\\partial_i\\partial_j+\\frac{1}{\\sqrt{|g|}}{\\partial_i(\\sqrt{|g|}g^{ij})}\\partial_j, |g|=\\det(g_{ij}).$$  \nThen we have $$\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}({B}_{L+5})}\\leq C\\lVert \\Delta_{ M_k}(u_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}({B}_{L+5})}\\rightarrow 0.$$\nHence, we get a new coordinate system $(u_k^1,\\cdots,u_k^n)$ and we discard the original coordinate system, and we use the same notation for tensors written in the new coordinate system, so in the new coordinate system we have\n$$\\lVert g_{k,ij}-\\delta_{ij} \\rVert_{C^{m+1,\\beta}(B_{L+3})}\\rightarrow 0.$$ \nNow we want to improve the convergence of $g_{k,ij}$     from elliptic equations.\nWe have a system of equations\n$$\\Delta_{M_k} g_{k,ij}+B_{ij}(g_k,\\partial g_k)=-2\\text{Ric}_{M_k,ij}. $$\nwhere $B_{ij}(g,\\partial g)$ are  polynormials of $g,\\partial g$ and are quadratic in $\\partial g$.\nFrom $W^{m+2,p}$ estimates, Morrey embeddings, and $$|\\nabla^l \\text{Ric}_{ M_k}|\\rightarrow 0, 0\\leq l\\leq m,$$\nwe have for $1\\leq i,j\\leq n$\n\n\\begin{equation}\\label{harmonic coordinate estimates for use of lower dimensional}\n    \\begin{split}\n\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\alpha}( B_{L+2})}\\leq& C(\\norm{\\Delta_{ M_k} (g_{k,ij}-\\delta_{ij})}_{C^m( B_{L+3})}\\\\&+\\norm{g_{k,ij}-\\delta_{ij}}_{L^\\infty(B_{L+3})})\\rightarrow 0.\n    \\end{split}\n\\end{equation}\nHence we get a $(2(L+2), Q,m+1,\\alpha)$ harmonic coordinate chart centered at $p_k'$, with $d_{g_k}(p_k',p_k)\\rightarrow 0$ , then  $r_h^{m,\\alpha}(p_k,{g}_k,Q)\\geq 2(L+1)$ for large $k$, which is a contradiction. \n\n\\textbf{Case 2}\n$d_{{g}_k}(p_k,\\Sigma_k)\\leq K.$\n\nA subsequence $(M_k,{g_k},p_k)$ converges in pointed $C^{m+1,\\beta}$ sense to a complete Riemannian manifold with boundary $(M_\\infty,g_\\infty,p_\\infty)$ and $(\\partial M_k,{g}_k,q_k)$ converges in $C^{m+1,\\beta}$ sense to $(\\partial M_\\infty,g_{\\infty}|_{\\partial M_\\infty},q_\\infty)$, where $q_k\\in\\Sigma_k$ is the unique foot point of $p_k$ in $\\Sigma_k$. Then $\\text{Ric}_{M_\\infty}=0$, $\\text{Ric}_{\\partial M_\\infty}=0$, $H_\\infty=0$, $inj_{\\partial M_\\infty}=\\infty$, $i_{b,M_\\infty}=\\infty$. Hence\n$(\\partial M_\\infty, g_\\infty|_{\\partial M_\\infty})$ is isometric to $(\\mathbb{R}^{n-1}, g_{flat})$. By (\\ref{scalar curvatures and hypersurface}), we have $S_\\infty=0$. Then by Lemma $\\ref{flat totally geodesic}$, $(M_\\infty,g_\\infty)$ is a smooth Riemannian manifold with boundary and $Rm_{M_\\infty}=0$. Since also $i_{b,M_\\infty}=\\infty$, $(M_\\infty, g_\\infty)$ is a isometric to $(\\mathbb{R}^n_+,g_{flat})$.  \n\nHence for any $L>2K+10$, there exist a coordinate $\\varphi_{0,k}:B_{L+5}^+\\rightarrow U_k\\subset M_k$, $\\varphi_{0,k}(0)=q_k$, $\\varphi_{0,k}(\\tilde{B}_{L+5})=U_k\\cap\\partial M_k$ such that\n$$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\beta}(B_{L+5}^+)}\\rightarrow 0, 1\\leq i,j\\leq n.$$  \nFirst, we solve for functions $v_k^\\nu$,     $1\\leq\\nu\\leq n-1$, \n\\begin{equation}\n\\Delta_{\\partial M_k}v_k^\\nu=0 \\ {\\rm in} \\ \\tilde{B}_{L+5} , v_k^\\nu|_{\\partial \\tilde{B}_{L+5}}=x^\\nu ,\n\\end{equation}\nThen we have $$\\lVert v_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(\\tilde{B}_{L+5})}\\leq C\\lVert \\Delta_{\\partial M_k}(v_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}(\\tilde{B}_{L+5})}\\rightarrow 0.$$\nNext, we solve for $1\\leq\\nu\\leq n-1$,\n\\begin{equation}\n\\Delta_{M_k} u_k^\\nu=0  \\ {\\rm in} \\ B_{L+5}^+, u_k^\\nu|_{\\tilde{B}_{L+5}}=v_k^\\nu, u_k^\\nu|_{{\\partial^+B_{L+5}^+}}=x^\\nu.\n\\end{equation}\nNote that $\\partial B^+_{L+5}$ is not a $C^1$-boundary, but it satisfies exterior sphere condition, so we can solve the equations by Perron's method to get a unique solution  $u_k^\\nu \\in C^\\infty((B_{L+5}^+)^\\circ)\\cap C^0(\\overline{{B}_{L+5}^+}$).\nFrom definitions and the estimates above, we have\n$$\n\\lVert\\Delta_{M_k}(u_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}(B^+_{L+5})}\\rightarrow 0,$$  $$\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(\\tilde{B}_{L+5})}\\rightarrow 0,$$ $$\\lVert u_k^\\nu-x^\\nu\\rVert_{L^\\infty(\\partial B^+_{L+5})}\\rightarrow 0,$$\nthen by maximum principle, we have  $$\\lVert u_k^\\nu-x^\\nu\\rVert_{L^\\infty(B^+_{L+5})}\\rightarrow 0,$$ and by Schauder estimates\n\n\\begin{equation}\n\\begin{split}\n\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(B_{L+4}^+)}\\leq C(&\\lVert \\Delta_{M_k}(u_k^\\nu-x^\\nu)\\rVert_{C^{m,\\beta}(B_{L+5}^+)}+\\lVert u_k^\\nu-x^\\nu\\rVert_{L^\\infty(B_{L+5}^+)}\\\\ \n&+\\lVert u_k^\\nu-x^\\nu\\rVert_{C^{m+2,\\beta}(\\tilde{B}_{L+5})})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nNext, we construct $u_k^n$ by solving\n\n$$\\Delta_{M_k}u_k^n=0 \\ {\\rm in}\\ B^+_{L+5},u_k^n|_{\\partial B_{L+5}^+}=x^n.  $$ We have\n\n\\begin{equation}\n    \\begin{split}\n        \\lVert u_k^n-x^n\\rVert_{C^{m+2,\\beta}(B^+_{L+4})}\\leq& C(\\lVert \\Delta_{M_k}(u_k^n-x^n)\\rVert_{C^{m,\\beta}(B_{L+5}^+)}\\\\&+\\lVert u_k^n-x^n\\rVert_{L^\\infty(B_{L+5}^+)}) \\rightarrow 0.\n    \\end{split}\n\\end{equation}\nHence we get a new coordinate system $(u_k^1,\\cdots,u_k^n)$ and we discard the original coordinate system, and we use the same notation for tensors written in both coordinate systems, so in the new coordinate system we have\n\\begin{equation}\\label{lower order convergence harmonic coordinates}\n    \\lVert g_{k,ij}-\\delta_{ij} \\rVert_{C^{m+1,\\beta}(B_{L+3}^+)}\\rightarrow 0.\n\\end{equation}\nNow we want to improve the convergence of $g_{k,ij}$ from elliptic equations with Neumann boundary conditions.\nWe have equations\n\\begin{equation}\\label{Ricci boundary equation}\n\\Delta_{\\partial M_k} g_{k,ij}+\\tilde{B}_{ij}(g_k,\\partial g_k)=-2\\text{Ric}_{\\partial M_k,ij} \n\\end{equation}\n\\begin{equation}\\label{Ricci equation}\n\\Delta_{M_k} g_{k,ij}+B_{ij}(g_k,\\partial g_k)=-2\\text{Ric}_{M_k,ij} \n\\end{equation}\nFix $\\theta\\in (\\beta,1),  p=\\frac{n}{1-\\theta}$, from $W^{m+2,p}$ estimates, Morrey embeddings, and $$|\\nabla_{\\partial M_k}^l \\text{Ric}_{\\partial M_k}|\\rightarrow 0, 0\\leq l\\leq m,$$\nwe have for $1\\leq i,j\\leq n-1$,\n\\begin{equation}\\label{lower dimensional convergence harmonic coordinates}\n\\begin{split}\n\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(\\tilde B_{L+2.5}^+)}\\leq &C(\\norm{\\Delta_{\\partial M_k} (g_{k,ij}-\\delta_{ij})}_{C^m(\\tilde B_{L+3}^+)}\\\\&+\\norm{g_{k,ij}-\\delta_{ij}}_{L^\\infty(\\tilde B_{L+3}^+)})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nBy Theorem 8.33 in \\cite{gilbarg2015elliptic},\n\\begin{equation}\n\\begin{split}\n\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(B_{L+2}^+)}  \\leq & C(\\norm{\\Delta_{ M_k} (g_{k,ij}-\\delta_{ij})}_{C^m(\\tilde B_{L+2.5}^+)}\\\\ &+\\norm{g_{k,ij}-\\delta_{ij}}_{L^\\infty( B_{L+2.5}^+)}\\\\ &+\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(\\tilde B_{L+2.5}^+)})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nNote that\n\\begin{equation}\\label{Neumann boundary condition 1}\nN_kg_k^{nn}=-2(n-1)H_kg_k^{nn},\n\\end{equation}\n\\begin{equation}\\label{Neumann boundary condition 2}\nN_kg_k^{in}=-(n-1)H_kg_k^{in}+\\frac{1}{2\\sqrt{g_k^{nn}}}g_k^{ij}\\partial_j g_k^{nn},\n\\end{equation}\nwhere $N_k=\\frac{g_k^{jn}\\partial _j}{\\sqrt{g_k^{nn}}}$ is the unit normal vector of $\\partial M_k$, $1\\leq i\\leq n-1$ and $j$ sums from $1$ to $n$, then we have Neumann boundary conditions for (\\ref{Ricci equation}).  \nFor simplicity, assume for a while $m=0$. Since $$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{1,\\beta}(B_{L+2}^+)}\\rightarrow 0 ,$$ $$|\\text{Ric}_{M_k, ij}|_{C^0(B_{L+2}^+)}\\rightarrow 0, 1\\leq i,j\\leq n,$$ $$|H_k|_{C^1(\\tilde{B}_{L+2})}\\rightarrow 0,$$ we have\n\n$$ \\norm{\\Delta_{M_k} (g_k^{nn}-\\delta^{nn})}_{C^0(B_{L+2}^+)}\\rightarrow 0,   \\norm{N_kg_k^{nn}}_{C^1(\\tilde B_{L+2})}\\rightarrow 0,$$\n then by Morrey embeddings (together with extensions), and $W^{2,p}$ estimates for Neumann boundary problems (for example, see a priori estimate 2.3.1.1 in \\cite{grisvard2011elliptic}),\n\n\\begin{equation}\n\\begin{split}\n\\norm{g_k^{nn}-\\delta^{nn}}_{C^{1,\\theta}(B^+_{L+1.7})}\\leq & C \\norm{g_k^{nn}-\\delta^{nn}}_{W^{2,p}(B_{L+1.8}^+)}\\\\\n\\leq & C( \\norm{g_k^{nn}-\\delta^{nn}}_{L^{p}(B_{L+2}^+)}+ \\norm{\\Delta_{M_k} (g_k^{nn}-\\delta^{nn})}_{L^p(B_{L+2}^+)}\\\\ &+\\norm{N_kg_k^{nn}}_{W^{1-\\frac{1}{p},p}(\\tilde B_{L+2})}\\rightarrow 0.\n\\end{split}\n\\end{equation}\nNow for $1\\leq l\\leq n-1$, since $$\\norm{\\Delta_{M_k}( g_k^{ln}-\\delta^{ln})}_{C^0(B_{L+2}^+)}\\rightarrow 0,$$ \nand\n\\begin{equation}\n\\begin{split}\n\\norm{N_kg_k^{ln}}_{W^{1-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})} &\\leq C(\\norm{g_k^{nn}-\\delta^{nn}}_{W^{2-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})}+\\norm{ H_k}_{W^{1-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})})\\\\& \\leq C(\\norm{g_k^{nn}-\\delta^{nn}}_{W^{2,p}({B}^+_{L+1.7})}+\\norm{ H_k}_{C^1(\\tilde{B}_{L+1.7})})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nThen \n\\begin{equation}\n\\begin{split}\n\\norm{g_k^{ln}-\\delta^{ln}}_{C^{1,\\theta}(B_{L+1.1}^+)}\\leq & C\\norm{g_k^{ln}-\\delta^{ln}}_{W^{2,p}(B_{L+1.2}^+)}\\\\ \\leq &C(\\norm{g_k^{ln}-\\delta^{ln}}_{L^{p}(B_{L+1.5}^+)}+\\norm{\\Delta_{M_k}( g_k^{ln}-\\delta^{ln})}_{L^p(B_{L+1.5}^+)}\\\\  &+\\norm{N_kg_k^{ln}}_{W^{1-\\frac{1}{p},p}(\\tilde{B}_{L+1.5})})\\rightarrow 0.\n\\end{split}\n\\end{equation}\nHence $$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{1,\\theta}(B_{L+1.1}^+)}\\rightarrow 0,  1\\leq i,j\\leq n.$$\nFor general $m\\geq 1$,  take  $m$-th derivatives of (\\ref{Ricci equation}) and the Neumann boundary conditions (\\ref{Neumann boundary condition 1})(\\ref{Neumann boundary condition 2}),\nand note that $$[\\partial _i,N_k]=(\\frac{\\partial_ig_k^{jn}}{\\sqrt{g_k^{nn}}}-\\frac{g_k^{jn}\\partial_i g_k^{nn}}{2\\sqrt{g_k^{nn}}^3})\\partial_j,$$ so we get a system of second order elliptic equations with Neumann boundary conditions in $\\partial_\\gamma   g_k^{nn}$ and $\\partial_\\gamma g_k^{ln}$, $|\\gamma|=m, 1\\leq l\\leq m-1$, with other terms freezed. Apply the previous estimates in the case $m=0$ and use (\\ref{lower order convergence harmonic coordinates})(\\ref{lower dimensional convergence harmonic coordinates}),\nwe get\n$$\\norm{g_k^{nn}-\\delta^{nn}}_{C^{m+1,\\theta}(B_{L+1}^+)}\\rightarrow 0,$$\nand then for $1\\leq l\\leq n-1,$\n$$\\norm{g_k^{ln}-\\delta^{ln}}_{C^{m+1,\\theta}(B_{L+1}^+)}\\rightarrow 0,$$\nhence  $$\\norm{g_{k,ij}-\\delta_{ij}}_{C^{m+1,\\theta}(B_{L+1}^+)}\\rightarrow 0,  1\\leq i,j\\leq n$$\nIn particular, take $\\theta=\\alpha$, one can  we get a $(\\frac{L+1}{4}, Q,m+1,\\alpha)$ harmonic coordinate chart centered at $p_k'$ , with $d_{g_k}(p_k',p_k)\\rightarrow 0.$ Then  $r_h^{m+1,\\alpha}(p_k,{g}_k,Q)\\geq \\frac{L}{4}$ for large $k$, which is a contradiction.\n\n\n\\end{proof}\n\n\n\n\n\\begin{remark}\nNote that the case $m=0$ is also true, and one should be a little careful with the geometric arguments in the proof. Actually, the arguments in \\cite{anderson2004boundary} prove a $C_*^{m+2}$ harmonic radius lower bound. \n\\end{remark}\n\n\\begin{remark}\\label{global harmonic radius lower bounds for complete manifolds with boundary}\nThe proof also shows that if $M$ is complete, $i_b\\geq i_0$, $inj_M\\geq i_0$, $inj_{\\partial M}\\geq i_0$ and (\\ref{bounds 1 up to m})(\\ref{bounds 2 up to m}) hold, then for any $p\\in M$\n\\begin{equation}\\label{harmonic radius lower bound globally}\n     r^{m+1,\\alpha}_h(p,g,Q)\\geq r_0(i_0,\\Lambda,m,\\alpha,Q).\n \\end{equation}\n\n\\end{remark}\n\n\n\n\nThe following corollary is a version we will use often.\n\n\\begin{corollary}\\label{corollary that will be used often}\nLet $(M_i,g_i)$ be a sequence of complete Einstein manifold with boundary. Suppose $i_b\\geq i_0, inj_{\\partial M}\\geq i_0, |Rm|\\leq C, |S|\\leq C, |\\nabla_{\\partial M}^k Rm_{\\partial M}|\\leq C_k,|\\nabla_{\\partial M}^{k+1}H|\\leq C_k, k\\geq 0$, then for any $p_i\\in M_i$,  there exists some subsequence such that  $(M_i,g_i,p_i)$ converges in pointed Cheeger-Gromov sense.\n\\end{corollary}\n\n\n\n\\begin{proof}\nBy Proposition \\ref{volume noncollasping}, $|Rm|\\leq C$, $|S|\\leq C$, $i_b\\geq i_0$,  together imply volume lower bounds of interiors balls of some fixed radius near boundary, hence also gives an interior injectivity radius lower bound from the following lemma. Then use Remark \\ref{global harmonic radius lower bounds for complete manifolds with boundary} and Proposition \\ref{harmonic radius lower bound}.\n\\end{proof}\n\nThe following lemma is well-known, which is a qualitative version of Theorem 4.3 in \\cite{cheeger1982finite} and can also be easily proved by contradiction arguments.\n\n\\begin{lemma}\\label{well-know interior inj lower bound} Let $(M,g)$ be a Riemannian manifold, and $B(p,r)$ be a metric ball that has compact closure. Suppose  $$\\sup\\limits_{B(p,r)}|Rm|\\leq C, \\emph{vol}(B(p,r))\\geq v, $$then there exists $r_0>0$ depending on $n, C,v,r$ such that $\\exp_q:B_{r_0}(0)\\subset T_q M\\rightarrow B(q,r_0)\\subset M$ is a diffeomorphism for any $q\\in B(p,\\frac{r}{2})$.\n\\end{lemma}\n\n\n\n\n\n\n\\section{Convergence of hyperk\\\"ahler manifolds}\\label{section main proof}\n\n\\subsection{Curvature estimates near the boundary}\nThis section serves as a first step for the proof of our main theorem.\nFor an Einstein manifold with boundary, if the boundary intrinsic and extrinsic geometry are controlled well and $i_b$ is bounded from below, we hope to control the interior geometry within $i_b$. To the author's knowledge, we do not know any general statement. We will first state and prove a version we need, and then discuss some lemmas needed in the proof.\n\n\\begin{theorem}\\label{good boundary}\nLet $(M,g)$ be a complete hyperk\\\"ahler 4-manifold with compact boundary. Suppose $|S|\\leq C$,  $|\\nabla^j_{\\partial M} Rm_{\\partial M}|\\leq C_j, |\\nabla^{j+1}_{\\partial M} H|\\leq C_j,    j=0,1,\\cdots$, $inj_{\\partial M}\\geq i_0$, $i_b\\geq i_0$, $\\int_{M}|Rm|^2\\leq C$. Then for any $r_1<i_0$, there exists $C'>0$, depending on $C,C_j,i_0,r_1$,  such that $\\sup\\limits_{N_{r_1}(\\partial M,g)}|Rm|\\leq C'$.\n\n\\end{theorem}\n\n\n\\begin{proof}\nWithout loss of generality, assume $i_0=1$. Denote $\\alpha={r_1}$, $\\beta=\\frac{1}{4}(1-\\alpha)$.  Suppose the conclusion is not true, we have a sequence $(M_i,g_i)$ satisfying the conditions, but $$\\sup\\limits_{N_{\\alpha}(\\partial M_i,g_i)}|Rm_{g_i}|\\rightarrow\\infty.$$ Let  $p_i\\in N_{r_1}(\\partial M_i,g_i)$  achieves this supremum. \n\n\\textbf{Claim 1} There exists a subsequence such that \n\\begin{equation}\\label{Distance curvature}\nd_{g_i}(p_i,\\partial M_i)^2|Rm_{g_i}(p_i)|\\rightarrow\\infty.\n\\end{equation}\nIf this is not true, we have $\\sup_i d_{g_i}(p_i,\\partial M_i)^2 |Rm_{g_i}|(p_i)<\\infty$. Rescale $\\tilde{g}_i=|Rm_{g_i}(p_i)|g_i$, then $|Rm_{\\tilde{g}_i}(p_i)|=1$, and $|Rm_{\\tilde{g}_i}|\\leq 1$ in $N_{\\alpha|Rm(p_i)|^{\\frac{1}{2}}}(\\partial M_i,\\tilde{g}_i)$, and $\\sup_id_{\\tilde{g}_i}(p_i,\\partial M_i)<\\infty$, $ i_{b,\\tilde g_i}\\geq |Rm_{g_i}(p_i)|^{\\frac{1}{2}}$ for all $i$. Hence by Corollary \\ref{corollary that will be used often}, $(M_i,g_i,p_i)$ subconverges in pointed Cheeger-Gromov sense to $(M_\\infty, \\tilde{g}_\\infty,p_\\infty)$, which is a complete Ricci-flat 4-manifold with flat, totally geodesic boundary, hence must be flat by Lemma \\ref{flat totally geodesic}. This contradicts that $|Rm_{g_\\infty}(p_\\infty)|=1$ and proves Claim 1.\n\nNow rescale $g_i$ in another way, let $g_i'=d_{g_i}(p_i,\\partial M_i)^{-2}g_i$, so $d_{g_i'}(p_i,\\partial M)=1$.\nSince $d_{g_i}(p_i,\\partial M_i)\\leq \\alpha$, the rescaled metric $g_i'$ satisfies $i_{b,g_i'}\\geq \\alpha^{-1}$ as well as all other conditions of the assumptions of the theorem, but with different bounds, regardness of whether $d_{g_i}(p_i,\\partial M_i)$ is uniformly bounded from below or not. Moreover, (\\ref{Distance curvature}) is equivalent to $|Rm_{g_i'}(p_i)|\\rightarrow\\infty$.\n\nBy the $\\epsilon$-regularity Theorem \\ref{Cheeger-Tian}, there exists a universal constant $\\epsilon_0$ such that  for sufficiently large $i$, $$\\int_{B_{{g}_i'}(p_i,\\beta )}|Rm_{g_i'}|^2\\geq\\epsilon_0.$$ \n\n\n\\textbf{Claim 2} There exists a subsequence such that\n$$ \\sup\\limits_{ N_{\\alpha}(\\partial M_i,g_i')}|Rm_{g_i'}|\\rightarrow \\infty, $$\nIf not, we have\n$ \\sup\\limits_{ N_{\\alpha}(\\partial M_i,g_i')}|Rm_{g_i'}|\\leq C$. By Lemma \\ref{volume noncollasping} and Bishop-Gromov volume comparison, $\\text{vol}(B_{g_i'}(p_i,\\beta))\\geq v$. Since $B_{g_i'}(p_i,\\beta)\\subset N_{\\alpha^{-1}(\\alpha+\\beta)}(\\partial M_i,g_i')\\subset N_{\\alpha+\\beta}(\\partial M_i,g_i)$, and the last one is diffeomorphic to $\\partial M_i\\times [0,\\alpha+\\beta]$, we conclude that there is no $-2$ curve in $B_{g_i'}(p_i,\\beta)$. By Proposition \\ref{no bubble},\n$|Rm_{g_i'}(p_i)|$ is bounded, which is a contradiction to Claim 1 and finishes the proof of Claim 2.\n\n\n\n\nNow Claim 2 enables us to get by induction, for each fixed positive integer $N$, $N$ sequences of metrics $g_i^{(0)}=g_i,g_i^{(1)}=g_i',\\cdots, g_i^{(N)}$, and points $p_i^{(j)}\\in N_{\\alpha}(\\partial M_i,g_i^{(j)})$ for  $0\\leq j\\leq N-1$, $p_i^{(0)}=p_i$, such that for $0\\leq j\\leq N-1$, $p_i^{(j)}$ achieves the supremum of $|Rm_{g_i^{(j)}}|$ in $N_{\\alpha}(\\partial M_i,g_i^{(j)})$, and\n\n $$|Rm_{g_i^{(j)}}(p_i)|\\rightarrow\\infty,$$  $$d_{g_i^{(j+1)}}(p_i^{(j)},\\partial M_i)=1,$$\n$$\\int_{B_{{g}_i^{(j+1)}}(p_i^{(j)},\\beta)}|Rm_{g_i^{(j+1)}}|^2\\geq\\epsilon_0,$$ \n\n\n$$ B_{g_i^{(j+1)}}(p_i^{(j)},\\beta)\\subset N_{\\alpha^{-1}(\\alpha+\\beta)}(\\partial M_i,g_i^{(j+1)})\\subset N_{\\alpha+\\beta}(\\partial M_i,g_i^{(j)}) ,$$ \n $$B_{g_i^{(j+1)}}(p_i^{(j)},\\beta)\\cap N_{\\alpha+\\beta}(\\partial M_i,g_i^{(j+1)})=\\emptyset . $$               \nIt follows that for each fixed $i$, $B_{g_i^{(j+1)}}(p_i^{(j)},\\beta)$ does not interect each other for different $j$. Since $\\int_{M_i}|Rm_{g_i}|^2\\leq C$, we have $N\\epsilon_0\\leq C$. This is a contradiction, since $N$ can be any positive integer.\n\\end{proof}\n \n\n\\begin{remark}\nThis theorem is purely local. In fact, by slightly modifying the proof, we see that if the bounds in the assumptions hold in a metric cyclinder $C(B_{\\partial M}(p,r_0),0,r_1)$  such that $\\exp^\\perp$ maps $B_{\\partial M}(p,r_0)\\times [0,r_1)$ diffeomorphically onto it, and such that $B_{\\partial M}(p,r_0)$ has compact closure,  then we have curvature bounds in any interior metric cyclinder $C(B_{\\partial M}(p,r_0'),0,r_1')$ with fixed $r_0'<r_0,r_1'<r_1$.\n\\end{remark}\n\n\n\nThe following collapsing $\\epsilon$-regularity theorem is originally due to Cheeger-Tian in \\cite{cheeger2006curvature}. Recently, in the hyperk\\\"ahler case, \\cite{sun2021collapsing} gives a simple proof by a blow-up argument and studying complete collapsing limits of hyperk\\\"ahler manifolds with bounded curvature.\n\n\n\n\n\n\n\\begin{proposition}\\label{Cheeger-Tian} There exists $\\epsilon,c$ such that the following holds:\nLet $(M^4,g)$ be an Einstein 4-manifold, $|\\emph{Ric}|\\leq 3$, $r\\leq 1$, and $B(p,r)$ is a metric ball that has compact closure. If $$\\int_{B(p,r)}|Rm|^2\\leq\\epsilon,$$ then $$\\sup\\limits_{B(p,\\frac{r}{2})}|Rm|\\leq c r^{-2}.$$\n\\end{proposition}\n\n\nThe following proposition plays an important role. To avoid redundancy, we state it here without proving, but we will prove a more general version later, see Proposition \\ref{no bubble 2}.\n\n\n\\begin{proposition}\\label{no bubble}\nLet $(M,g)$ be an hyperk\\\"ahler 4-manifold. Suppose $B(p,5)$ has compact closure, $B(p,3)$\ncontains no $-2$ curve,  $$\\emph{vol}(B(p,1))\\geq v,$$\n$$\\int_{B(p,3)}|Rm|^2\\leq C.$$Then there exists $C'>0$, depending on $v,C$ such that $$\\sup_{B(p,1)}|Rm|\\leq C'.$$\n\\end{proposition}\n\n\n\n\nThe following lemma originally dates back to Koiso in \\cite{koiso1981hypersurfaces}, \nand is an incredibly special case of the result in \\cite{biquard:hal-02928859}\\cite{anderson2008unique}.\nSince it plays an important role throughout the paper, we provide a detailed proof here following \\cite{koiso1981hypersurfaces}.\n\n\n\n\n\n\\begin{lemma}\\label{flat totally geodesic}\nLet $(M,g)$ be a connected  $C^2$ Riemannian manifold with boundary. Suppose $\\emph{Ric}_M=0$ and for some open boundary portion $T$, $S|_T=0$ and $Rm_{\\partial M}|_T=0$, then $g$ is smooth and $Rm_M=0$.\n\\end{lemma}\n\n\\begin{proof}\nFix any point $p$ in $T$. First, we show $g$ can be extended across the boundary near $p$.\nChoose a semi-geodesic coordinate system $(x^1,\\cdots, x^n)$ near $p$, with $\\partial M$ identified with $\\{x^n=0\\}$, and interior identified with $\\{x^n>0\\}$, $\\nabla x^n=\\partial_{ x^n}$, and $g_{ij}(x',0)=\\delta_{ij}, 1\\leq i,j\\leq n-1$, where $x'=(x^1,\\cdots, x^{n-1})$.  We extend the metric tensor $g$ by reflection across $\\{x^n=0\\}$, i.e., set $g_{ij}(x',x^n)=g_{ij}(x',-x^n), 1\\leq i,j\\leq n$. Then $g\\in C^0(B)\\cap C^2(B^+)$, where $B^+$ is the boundary coordinate ball and $B$ is its extension after reflection. We need to show $g\\in C^2(B)$. In fact, $S=0$ is equivalent to $\\frac{\\partial g_{ij}}{\\partial x^n_+}(x',0)=0, 1\\leq i,j\\leq n-1,$ then we also have $\\frac{\\partial g_{ij}}{\\partial x^n_-}(x',0)=-\\frac{\\partial g_{ij}}{\\partial x^n_+}(x',0)=0,$ hence  $\\frac{\\partial g_{ij}}{\\partial x^n}(x',0)=0$ and $g_{ij}\\in C^1(B)$. Since $g\\in C^2(B^+)$, we have for $1\\leq l\\leq n-1$,\n $$\\frac{\\partial }{\\partial x^n_+}\\frac{\\partial g_{ij}}{\\partial x^l}(x',0)=\\frac{\\partial }{\\partial x^l}\\frac{\\partial g_{ij}}{\\partial x^n_+}(x',0)=\\frac{\\partial }{\\partial x^l}0=0,$$ $$\\frac{\\partial }{\\partial x^n_-}\\frac{\\partial g_{ij}}{\\partial x^l}(x',0)=-\\frac{\\partial }{\\partial x^n_+}\\frac{\\partial g_{ij}}{\\partial x^l}(x',0)=0.$$ \n \\begin{equation}\n \\begin{split}\n      \\frac{\\partial}{\\partial x^n_-}\\frac{\\partial g_{ij}}{\\partial x^n}(x',0)&=\\lim\\limits_{x^n\\rightarrow 0^-} \\frac{1}{x^n}\\frac{\\partial g_{ij}}{\\partial x^n}(x',-x^n)\\\\ &=\\lim\\limits_{x^n\\rightarrow 0^+}\\frac{1}{x^n}\\frac{\\partial g_{ij}}{\\partial x^n}(x',x^n)=\\frac{\\partial}{\\partial x^n_+}\\frac{\\partial g_{ij}}{\\partial x_n}(x',0).\n \\end{split}\n \\end{equation}\n  Hence $g_{ij}\\in C^2(B)$. Finally, we have $g_{nn}=1,g_{ln}=g_{nl}=0, 1\\leq l\\leq n-1$, hence $g\\in C^2(B)$.\n\n\nBy elliptic regularity, all harmonic coordinate charts in $B$ give rise to a real analytic structure in $B$ such that $g$ is real analytic. Hence if $t$ is a distance function such that $\\partial M$ is defined by $t^{-1}(0)$ near $p$, then $t$ is real analytic near $p$. Choose a real analytic coordinate $(z,t)$ near $p$. Since $t^{-1}(0)$ is totally geodesic, $\\frac{\\partial g}{\\partial t}(z,0) =0$. The evolution equation (\\ref{evolution equation for 2nd form}) is equivalent to the second order PDE\n\\begin{equation}\n\\frac{\\partial^2 g}{\\partial t^2}=2 ric \\ g-\\frac{1}{2}tr_g(\\frac{\\partial g}{\\partial t})\\frac{\\partial g}{\\partial t}+(\\frac{\\partial g}{\\partial t})^2,\n\\end{equation}\n where $ric\\ g$ is the Ricci tensor of level sets of $t$. By the uniqueness part of Cauchy-Kovalevskaya theorem, we know $g(z,t)=g(z,0)$. Hence $Rm_M=0$ near $p$. Since $Rm_M$ is real analytic in the interior of $M$, $Rm_M=0$ in $M$.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Convergence of hyperk\\\"ahler metrics}\n\n\\begin{theorem}\\label{main theorem hyperkahler metric}\nLet $(X_i,g_i)$ be a sequence of compact, connected hyperk\\\"ahler 4-manifold with boundary, suppose on $\\partial X_i$, we have\n$$H_i\\geq H_0>0, |S_i|\\leq C, |\\nabla_{\\partial X_i}^{j+1} H|\\leq C_j,$$   $$inj_{\\partial X_i}\\geq i_0, \\emph{diam}_{g_i|_{\\partial X_i}}(\\partial X_i)\\leq C,|\\nabla^j_{\\partial X_i}Rm_{\\partial X_i}|\\leq C_j,\\forall j\\geq 0, $$ \nand $\\chi(X_i)\\leq C$.\nAssuming there exists no $-2$ curve on $X_i$, \nthen there exists a subsequence such that $(X_i,g_i)$ converges in Cheeger-Gromov sense to a compact, connected hyperk\\\"ahler 4-manifold with boundary $(X_\\infty, g_\\infty).$\n\\end{theorem}\n\n\\begin{remark}\nBy Chern-Gauss-Bonnet formula, our assumptions imply $$\\int_{X_i}|Rm|^2\\leq C.$$ Note that for a compact connected Einstein 4-manifold $(M,g)$ with boundary, the Chern-Gauss-Bonnet formula says\n$$\\frac{1}{8\\pi^2}\\int_M|Rm|^2=\\chi(M)-\\frac{1}{2\\pi^2}\\int_{\\partial M}\\prod\\limits_{i=1}^3\\lambda_i-\\frac{1}{8\\pi^2}\\int_{\\partial M}\\sum\\limits_{\\sigma\\in S_3}K_{\\sigma_1\\sigma_2}\\lambda_{\\sigma_3}. $$\nHere $\\lambda_1,\\lambda_2,\\lambda_3$ are eigenvalues of the shape operator $S$ of $\\partial M$, Let $e_i$ be eigenvectors of eigenvalue $\\lambda_i$ such that $\\{e_1,e_2,e_3\\}$ is an orthonormal basis, then $K_{ij}=\\sec(e_i,e_j) $. See for example (1.16) in \\cite{Anderson2000L2CA}.\n\\end{remark}\n\n\n\\begin{remark}\nIf we drop the condition $\\text{diam}_{g_i|_{\\partial X_i}}(\\partial X_i)\\leq C$, and replace $H_i\\geq H_0>0$ by $H_i>0$, $\\chi(X_i)\\leq C$ by $\\int_{X_i}|Rm_{g_i}|^2\\leq C$, then for any point $p_i\\in X_i$, a subsequence of $(X_i,g_i,p_i)$ converges in pointed Cheeger-Gromov sense to a complete, connected hyperk\\\"ahler 4-manifolds with nonempty or empty boundary $(X_\\infty,g_\\infty,p_\\infty)$, depending on whether the distance of $p_i$ to boundary is bounded or not. \n\\end{remark}\n\n\\begin{remark}\\label{positive mean curvature essential}\nThe positive mean curvature condition is necessary. The following counterexample is natural and was observed by Donaldson in \\cite{donaldson2018remarks}. Consider the standard unit ball $B^4$  inside Euclidean $\\mathbb{R}^4$, ``squeeze'' the ball such that the north pole and the south pole of the boundary $S^3$ comes together, so we get a sequence of embedded $B^4$ in $\\mathbb{R}^4$ converging in Hausdorff sense to a limit homeomorphic to the wedge sum of two $B^4$, whose boundary is an immersed $S^3$ intersecting itself at one point. For this sequence, all other assumptions are satisfied. Slightly modifying the process, one can also have a sequence of $B^4$ of dumbbell shape such that the middle cyclinder $B^3\\times [0,1]$ collapses to $[0,1]$, then they have a Hausdorff limit which is homeomorphic to two $B^4$ joint by a line segment.\n\nIn these types of examples, the curvatures are uniformly bounded, and the global volume are non-collapsing before taking the limit.\n\n\n\\end{remark}\n\n\n\\begin{remark}\\label{no -2 curve essential}\n\nIf we allow $-2$ curves and do not assume the positive mean curvature condition, something worse will happen: consider the Kummer construction. Let $T^4\/\\mathbb{Z}_2$ be the flat 4-orbifold with 16 singularities, remove small neighborhoods of the 16 singularities, and glue in 16 copies of $T^*S^2$. By varing the sizes of these glue-in regions and perturbing the metrics, we get a sequence of hyperk\\\"ahler 4-manifolds $(M_i,h_i)$, each of which is diffeomorphic to a $K3$ surface, such that $(M_i,h_i)$ converges in Gromov-Hausdorff sense to $T^4\/\\mathbb{Z}_2$, and converges in Cheeger-Gromov sense away from these 16 singularities. Now let $T^3\/\\mathbb{Z}_2\\subset T^4\/\\mathbb{Z}_2$ be the flat 3-orbifold, such that the last coordinate equal to 0. Let $\\tilde{X}_i\\subset T^4\/\\mathbb{Z}_2$  be the closure of the tubular neighborhood of $T^3\/\\mathbb{Z}_2$ of width $i^{-1}$, then $\\partial \\tilde{X}_i$ is connected, totally geodesic, isometric to the same flat $T^3$. Now for each $i$, choose $n(i)$ large enough such that one can find  $X_i\\subset M_{n(i)}$ which are compact domains with smooth boundary,  $d_{GH}(X_i,\\tilde{X}_i)\\rightarrow 0$, $|\\nabla_{\\partial X_i}^j S_{\\partial X_i}|\\rightarrow 0 $, $\\forall j\\geq 0$, $\\partial X_i$ converges in Cheeger-Gromov sense to $\\partial \\tilde X_i$. Let $g_i=h_{n(i)}|_{ X_i}$, then $(X_i,g_i)$ converges in Gromov-Hausdorff sense to flat $T^3\/\\mathbb{Z}_2$. In this case, $\\text{vol}(X_i)\\rightarrow 0$, $\\sup\\limits_{X_i}|Rm_{g_i}|\\rightarrow\\infty$.\n\nNote that in this example $\\partial \\tilde{X}_i$ cannot be perturbed in flat $T^4\/\\mathbb{Z}_2$ to have positive mean curvature. In fact,\ntake a small tubular neighborhood of $\\partial \\tilde{X}_i$ of width less than $i^{-1}$, whose boundary has two totally geodesic connected components $T_1,T_2$. Suppose $\\partial \\tilde{X}_i$ can be perturbed to $T'$ such that its mean curvature has a strict sign, say that its mean curvature vector points towards $T_1$. Then $T', T_1$ bound a region $W$. By \\cite{donaldson2018remarks} Proposition 7, $T'$ and $T_1$ are isometric, so $T'$ is totally geodesic, which is a contradiction.  \n\n\\end{remark}\nThe major step of the proof in \\ref{main theorem hyperkahler metric} is to show that these Riemannian manifolds have a nice neighborhood of definite size. We show that the boundary injectivity radius has a lower bound, so that the interior geometry within the boundary injectivity radius is nicely controlled by Theorem \\ref{good boundary}.\n\n\\begin{proposition}\\label{global inj lower bound}\nThere exists $i_1>0$, depending on the constants in Theorem \\ref{main theorem hyperkahler metric} such that    $i_{b,g_i}\\geq i_1.$\n\\end{proposition}\n\n\\begin{proof}\nSuppose not, we have a subsequence of hyperk\\\"ahler metrics $g_i$,  with $i_{b,{g}_i}\\rightarrow 0$. Rescale the metric $\\tilde g_i=i_{b,g_i}^{-2}{g}_i$, then $i_{b,\\tilde g_i}=1$. For any point $p_i\\in \\partial X_i$, the restriction metric $(\\partial X_i,\\tilde{g}_i|_{\\partial X_i},p_i)$ converges in pointed Cheeger-Gromov sense to  flat $\\mathbb{R}^3$, $|S_{\\tilde{g}_i}|\\rightarrow 0$ and $|\\nabla_{\\partial X_i}^{j+1} H_i|\\rightarrow 0$ uniformly on $\\partial X_i$. Consider $\\sup\\limits_{B_{\\tilde g_i}(p_i,4)}|Rm_{\\tilde g_i}|$. We have two cases \n\n\\textbf{Case 1} $\\sup\\limits_{B_{\\tilde g_i}(p_i,4)}|Rm_{\\tilde g_i}|\\leq C.$\\\\\nWe have a subsequence  $(B_{\\tilde g_i}(p_i,3),\\tilde g_i)$ converges in Cheeger-Gromov sense to a Riemannian manifold with boundary  $(B_\\infty,g_\\infty)$, so $(B_\\infty,g_\\infty)$ is Ricci flat, and all boundary components are flat, totally geodesic. By Lemma \\ref{flat totally geodesic}, $(B_\\infty,g_\\infty)$ is flat. We need to choose good points $p_i$ to lead to a contradiction. In fact, by Proposition \\ref{existence of focal points}, there exists $p_i\\in\\partial X_i$ such that $\\gamma_{p_i}(1)$ is a focal point along $\\gamma_{p_i}$. Let $p_\\infty$ be the limit of $p_i$. By Proposition \\ref{pass to limit}, we get a limit geodesic $\\gamma_{p_\\infty}: [0,1]\\rightarrow B_\\infty$, such that $\\gamma_{p_\\infty}(1)$ is a focal point along $\\gamma_{p_\\infty}$, contradiction. \n\n\\textbf{Case 2} For some subsequence we have $\\sup\\limits_{B_{\\tilde g_i}(p_i,4)}|Rm_{\\tilde g_i}| \\rightarrow\\infty$.\\\\\nThen we can find points $q_i\\in B_{\\tilde{g}_i}(p_i,4)$ such that $|Rm_{\\tilde g_i}(q_i)|\\rightarrow\\infty$. By Theorem \\ref{good boundary}, we have $d_{\\tilde g_i}(q_i,\\partial X_i)\\geq \\frac{1}{2}$ for large $i$. By Theorem \\ref{good boundary}, Proposition \\ref{volume noncollasping}, and the fact  $d_{\\tilde{g}_i}(q_i,\\partial X_i)\\leq 4$, we have  $\\text{vol}_{\\tilde{g}_i}(B_{\\tilde{g}_i}(q_i,\\frac{r_0}{2}))\\geq v_0$ for some $r_0,v_0$. Since there is no $-2$ curve in $X_i$, then by Proposition \\ref{no bubble}, $\\sup_{B_{\\tilde{g}_i}(q_i,\\frac{r_0}{10})}|Rm_{\\tilde{g}_i}|\\leq C$, which is a contradiction.\n\n\\end{proof}\n\n\n\n\n\n\nThen we can finish the proof of Theorem \\ref{main theorem hyperkahler metric} as follows: by Theorem \\ref{good boundary}, $|Rm_{g_i}|$ is uniformly bounded in $N_{\\frac{i_1}{2}}(\\partial X_i,g_i)$. By Proposition $\\ref{volume noncollasping}$, there exist $r_1<\\frac{i_1}{10},v_1$, such that $\\text{vol}_{g_i}(B_{g_i}(p,r_1))\\geq v_1$ for any $p$ with $d_{g_i}(p,\\partial X_i)=2r_1$.  By Proposition \\ref{diam}, $\\sup\\limits_{q\\in X_i}d_{g_i}(q,\\partial X_i)\\leq 3H_0^{-1}$, hence $\\text{diam} (X_i,g_i)\\leq C$. Then from Bishop-Gromov volume comparison,  we know $\\text{vol}_{g_i}(B_{g_i}(p,r_1))\\geq v_2$ for any $p$ with $d_{g_i}(p,\\partial X_i)\\geq 5r_1$.  By Proposition $\\ref{no bubble}$, for these $p$,\n$|Rm_{g_i}(p)|$ is uniformly bounded, with the bound independent of $i$ and $p$. Hence $\\sup_X |Rm_{g_i}|$ is uniformly bounded. Then by Corollary \\ref{corollary that will be used often},\n a subsequence $(X_i,g_i)$ converges in Cheeger-Gromov sense to a smooth Riemannian manifold with boundary $(X,g)$, so $g$ is a hyperk\\\"ahler metric.\n\n\n\n\\begin{remark} One can also directly prove Theorem \\ref{main theorem hyperkahler metric} by rescaling the maximum curvature norm to be 1. Suppose it is achieved at $p_i$. Then from Corollary \\ref{Ric positive, mean positive, inj lower bound}, we know $i_b\\geq i_0$ for the rescaled metrics. If now $d(p_i,\\partial X_i)$ is bounded, then the pointed Riemannian manifolds converge in pointed Cheeger-Gromov sense to a Ricci-flat manifold with flat and totally geodesic boundary, hence the limit is flat, contradition. Otherwise, for a subsequence $d(p_i,\\partial X_i)\\rightarrow\\infty$, then we rescale the metrics again such that this distance is 1. However, it is unknown whether the curvature is bounded in a fixed size neighborhood of the boundary in this scale. Using the idea of Theorem \\ref{good boundary}, we can keep rescaling such that this will happen at some stage, for possibly different points $p_i'$, then get a contradiction. The contradiction is the same the one in Case 2 in Proposition \\ref{global inj lower bound}, since volume lower bounds can be passed within finite distance, so do the curvature bounds by Proposition \\ref{no bubble}. Alternatively, one can prove Theorem \\ref{main theorem hyperkahler metric} by rescaling the harmonic radius, and all these methods turn out to be equivalent eventually.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Convergence of hyperk\\\"ahler triples}\\label{moduli space}\n\nNow we turn to the setting of \\ref{convergence of triples}. ${X}$ will denote a compact oriented 4-manifold with boundary $\\partial X=Y$. Let $\\mathcal{N}^+$ be the set of closed framings $\\bm{\\gamma}=(\\gamma_1,\\gamma_2,\\gamma_3)$ on $Y$  that satisfies the ``positive mean curvature'' condition\n\n\\begin{equation}\\label{mean positive}\n\\sum\\limits_{i=1}^3 \\langle\\gamma_i,d(*_{\\bm\\gamma}\\gamma_i)\\rangle_{\\bm\\gamma}>0.\n\\end{equation}\nLet $\\mathcal{M}^+$ be the set of smooth hyperk\\\"ahler triples $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ whose restriction to the boundary lies in $\\mathcal{N}^+$, so we have a restriction map $p_0:\\mathcal{M}^+\\rightarrow\\mathcal{N}^+$, which induce a map\n\\begin{equation}\\label{map between moduli space}\n    p:\\mathcal{M}^+\/\\mathcal{G}_X\\rightarrow\\mathcal{N}^+\/\\mathcal{G}_Y.\n\\end{equation}\n  Here $\\mathcal{M}^+$, $\\mathcal{N}^+$ are equipped with Fr\\'echet topology defined by smooth convergence, $\\mathcal{G}_X,\\mathcal{G}_Y$ are orientation preserving diffeomorphism groups of $X,Y$, respectively. It is obvious that $p_0$, $p$ are continuous. \nThen the compactness part of Theorem \\ref{convergence of triples} is equivalent to:\n\\begin{proposition}\nWhen there is no $-2$ curve in $X$, the map $p:\\mathcal{M}^+\/\\mathcal{G}_X\\rightarrow\\mathcal{N}^+\/\\mathcal{G}_Y$   is proper.\n\\end{proposition}\n\n\\begin{proof}\nSuppose for some $\\phi_i\\in\\mathcal{G}_Y$, and $\\bm\\gamma_i\\in\\mathcal{N}^+$,we have          $\\phi_i^*{\\bm\\gamma_i}\\rightarrow \\bm\\gamma\\in \\mathcal{N}^+$ and there exist $\\bm\\omega_i\\in\\mathcal{M}^+$ with $\\bm\\omega_i|_Y=\\bm\\gamma_i$, we want to show there exists $\\psi_i\\in\\mathcal{G}_X$ and $\\bm\\omega\\in\\mathcal{M}^+$  such that $\\psi_i^*\\bm\\omega_i\\rightarrow\\bm\\omega$. Let $g_i$ be the Riemannian metric defined by $\\bm\\omega_i$. By the assumptions, we have a uniform positive lower bound for the mean curvature $H_i$ of $Y$ of the metric $g_{i}=g_{\\bm\\omega_i}$, and bounds for $|\\nabla_{Y}^l S_i|$ for all $l\\geq 0$, and $(Y,g_i|_{Y})$ converges in Cheeger-Gromov sense. Hence $(X,g_i)$ satisfies all conditions in Theorem \\ref{main theorem hyperkahler metric}. Then for a subsequence, there exists diffeomorphism $\\psi_i: X\\rightarrow X$ such that $\\psi_i^*g_i\\rightarrow g$ smoothly as tensors. One can assume $\\psi_i$ is orientation preserving, otherwise, for a subsequence, compose them with a fixed orientataion reversing diffeomorphism of $X$. Since $|\\psi_i^*\\bm\\omega_i|^2=3$, $\\psi_i^*\\bm\\omega_i$ are parallel, we conclude that for some subsequence $\\psi_i^*\\bm\\omega_i\\rightarrow\\bm\\omega$ smoothly, and $\\bm\\omega\\in\\mathcal{M}^+$.\n\n\\end{proof}\n\n\n\n\n\\subsection{Enhancements}\n\nFor the proof of the compactness part of Theorem \\ref{convergence of triples enhancements}, one only needs the following proposition in place of Proposition \\ref{no bubble}. Then we argue in the same way as the proof of Theorem \\ref{convergence of triples}. \n\n\n\n\\begin{proposition}\\label{no bubble 2}\nLet $(M,g,\\bm\\omega)$ be a hyperk\\\"ahler 4-manifold. Suppose $B(p,5)$ has compact closure, and for any $-2$ curve $\\Sigma$ in $B(p,3)$\n, $$\\Big{|}\\int_{\\Sigma}\\bm\\omega\\Big{|}\\geq a>0,$$  $$\\emph{vol}(B(p,1))\\geq v,$$ \n$$\\int_{B(p,3)}|Rm|^2\\leq C. $$ Then there exists $C'>0$, depending on $a,v,C$ such that $$\\sup_{B(p,1)}|Rm|\\leq C'.$$\n\\end{proposition}\n\n\n\\begin{proof}\n\nFor all $q\\in B(p,2)$, by volume comparison, we have $$\\text{vol}(B(q,1))\\geq 3^{-4} \\text{vol}(B(q,3))\\geq 3^{-4}\\text{vol}(B(p,1))\\geq 3^{-4} v.$$\nSuppose the conclusion is not true, then we have a sequence $(M_i,g_i,p_i)$ satisfies the conditions, but\n there exists $q_i'\\in B(p_i,1)$ with $|Rm_{g_i}(q_i')|\\rightarrow\\infty$. By the following Lemma \\ref{point selection}, we can find points $q_i\\in B(p_i,2)$ such that $|Rm_{g_i}(q_i)|\\geq |Rm_{g_i}(q_i')|$, and $$\\sup\\limits_{B_{g_i}(q_i,|Rm_{g_i}(q_i')|^{\\frac{1}{2}}|Rm_{g_i}(q_i)|^{-\\frac{1}{2}})}|Rm_{g_i}|\\leq 4|Rm_{g_i}(q_i)|.$$  Rescale the metric $\\tilde{g}_i=|Rm_{g_i}(q_i)|g_i$, $\\tilde{\\bm\\omega}_i=|Rm_{g_i}(q_i)|\\bm\\omega_i$, so $\\tilde{\\bm\\omega}_i$ defines $\\tilde{g}_i$. Then we have $$\\sup\\limits_{B_{\\tilde{g}_i}(q_i,|Rm_{g_i}(q_i')|^{\\frac{1}{2}})}|Rm_{\\tilde{g}_i}|\\leq 4 , $$  $$|Rm_{\\tilde{g}_i}(q_i)|=1,$$ $$ \\text{vol}_{\\tilde{g}_i}(B_{\\tilde{g}_i}(q_i,r))\\geq 3^{-4}vr^n,\\forall r\\leq|Rm_{g_i}(q_i)|^{\\frac{1}{2}},$$ and\n$$\\int_{B_{\\tilde{g}_i}(q_i, |Rm_{g_i}(q_i)|^{\\frac{1}{2}})}|Rm_{\\tilde{g}_i}|^2\\leq \\int_{B_{\\tilde{g}_i}(p_i, 3|Rm_{g_i}(q_i)|^{\\frac{1}{2}})}|Rm_{\\tilde{g}_i}|^2\\leq C.$$\nHence, for a subsequence, $(M_i,\\tilde{g}_i,q_i)$ converges in Cheeger-Gromov topology to a complete non-flat hyperk\\\"ahler 4-manifold  $(M_\\infty,g_\\infty,q_\\infty)$ with maximum volume growth and $$\\int_{M_\\infty}|Rm_{g_\\infty}|^2\\leq C.$$\nSince $\\tilde{\\bm\\omega}_i$ are parallel, and $|\\tilde{\\bm\\omega}_i|^2=6$, $\\tilde{\\bm\\omega}_i$ also subconverges to a hyperk\\\"ahler triple $\\bm\\omega_\\infty$ that defines $g_\\infty$. By \\cite{bando1989construction}, $(M_\\infty,g_\\infty)$ is a hyperk\\\"ahler ALE space of order 4. Hence, from Kronheimer's classification \\cite{kronheimer1989construction}\\cite{kronheimer1989torelli}, there exists a $-2$ curve $C_\\infty$ in $B_{g_\\infty}(q_\\infty, R)$ for some $R>0$\nsuch that $|\\int_{C_\\infty}\\bm\\omega_{\\infty}|\\neq 0$.\nLet $\\phi_i:B_{g_\\infty}(q_\\infty,R)\\rightarrow V_i$ be diffeomorphisms, such that $\\phi_i^*\\tilde{\\bm\\omega}_i\\rightarrow \\bm\\omega_\\infty$ smoothly. Let $C_i=(\\phi_i)_*C_\\infty$, then $C_i$ is a $-2$ curve in $B_{g_i}(p_i,3)$ for large $i$ and $$\\int_{C_i} \\bm\\omega_i=|Rm_{g_i}(q_i)|^{-1}\\int_{C_i}\\tilde{\\bm\\omega}_i=|Rm_{g_i}(q_i)|^{-1}\\int_{C_\\infty} \\phi_i^*\\tilde{\\bm\\omega}_i \\rightarrow 0, $$\nwhich contradicts our assumption.\n\n\\end{proof}\n\n\nThe following point selection lemma is well-known and elementary. \n\n\\begin{lemma}\\label{point selection}\nLet $(M,g)$ be a Riemannian manifold. Suppose $\\sup\\limits_{B(p,2)}|Rm|<\\infty$, $|Rm(p)|\\neq 0$. Then there exists a point $q\\in B(p,2)$ such that $|Rm(q)|\\geq |Rm(p)|$, and $\\sup\\limits_{B(q,{|Rm(p)|}^{\\frac{1}{2}}{|Rm(q)|^{-\\frac{1}{2}}})}|Rm|\\leq  4|Rm(q)|.$  \n\\end{lemma}\n\n\\begin{proof}\nIf this is not true, let $A=|Rm(p)|^{\\frac{1}{2}}$, then there exist $q_1\\in B(p,2)$ such that $d(q_1,p)< 1$ and $|Rm(q_1)|> 4|Rm(p)|=4A^2$. By induction, we can find a sequence of points $q_0=p, q_1,q_2,\\cdots$ with $d(q_{j+1},q_j)< |Rm(q_{j})|^{-\\frac{1}{2}}A$, $d(q_{j+1},p)< 2-2^{-j}$ and $|Rm(q_{j+1})|>  4^{j+1} A^2$. This is an obvious contradiction since $\\sup\\limits_{B(p,2)}|Rm|<\\infty.$\n\\end{proof}\n\n\n\\subsection{Uniqueness}\\label{uniqueness 1}\n\n\nGiven the compactness results, it is natural to ask whether a subsequential limit $\\bm\\omega$ is unique(up to a diffeomorphism) in Theorem \\ref{convergence of triples}, Theorem \\ref{convergence of triples enhancements}. The answer is yes. In fact, both\\cite{biquard:hal-02928859} and \\cite{anderson2008unique} proved a\nunique continuation theorem for Einstein metrics with prescribed boundary metric and second fundamental form, which implies the following:\n\n\\begin{proposition}\\label{local uniqueness hyperkahler triple}\nLet $X$ be a connected oriented 4-manifold with boundary. Suppose $\\bm\\omega_1,\\bm\\omega_2$ are two smooth hyperk\\\"ahler triples on $X$. Suppose $\\bm\\omega_1|_{\\partial X}=\\bm\\omega_2|_{\\partial X}$ , then in geodesic gauges of $g_{\\bm\\omega_1}$, $g_{\\bm\\omega_2}$, we have\n$\\bm\\omega_1=\\bm\\omega_2$ near $\\partial X$. \n\\end{proposition}\n\n\\begin{proof} \nWe have $g_{\\bm\\omega_1}|_{\\partial X}=g_{\\bm\\omega_2}|_{\\partial X}$ and $II_{g_{\\bm\\omega_1}}=II_{g_{\\bm\\omega_2}}$ on $\\partial X$. By \\cite{biquard:hal-02928859} Theorem 4, in geodesic gauges, we have $g_{\\bm\\omega_1}=g_{\\bm\\omega_2}$. Since $\\nabla^{g_{\\bm\\omega_1}} |\\bm\\omega_1-\\bm\\omega_2|_{g_{\\bm\\omega_1}}^2=0$, and $\\bm\\omega_1=\\bm\\omega_2$ at one point, we have $\\bm\\omega_1=\\bm\\omega_2$ everywhere near $\\partial X$.\n\\end{proof}\n\n\n\nNow the following global uniqueness result follows from an analytic continuation argument (See \\cite{kobayashi1963foundations}  Chapter VI, Section 6): \n\n\n\n\\begin{theorem}\\label{unique continuation hyperkahler}\nLet $X$ be a connected 4-manifold with boundary,  $\\pi_1(X,\\partial X)=0$. Suppose $\\bm\\omega_1,\\bm\\omega_2$ are two smooth hyperk\\\"ahler triples on $X$, and $\\phi_0:\\partial X\\rightarrow\\partial X$ is a diffeomorphism, such that $\\bm\\omega_1|_{\\partial X}=\\phi_0^*(\\bm\\omega_2|_{\\partial X})$, then there exists a diffeomorphism $\\phi:X\\rightarrow X$,   $\\phi|_{\\partial X}=\\phi_0$ such that  $\\bm\\omega_1=\\phi^*\\bm\\omega_2$ on $X$.\n\\end{theorem}\n\n\n\\begin{remark} This theorem implies that\nthe map $p:\\mathcal{M}^+\/\\mathcal{G}_X\\rightarrow \\mathcal{N}^+\/\\mathcal{G}_Y$ defined in subsection \\ref{moduli space} is injective, provided that $\\mathcal{M}^+$ is nonempty.\n\\end{remark}\n \n \n\\begin{proof}\nBy the previous proposition, there exists a collar neighborhood $U$ of $\\partial X$ and a diffeomorphism $\\phi_1: U\\rightarrow V\\subset X$ such that $g_{\\bm\\omega_1}=\\phi_1^*g_{\\bm\\omega_2}$, $\\phi_1|_{\\partial X}=\\phi_0$. Since $g_{\\bm\\omega_1},g_{\\bm\\omega_2}$ are real analytic, $\\phi_1$ is real analytic.  Fix $p_0\\in U$ and a small neighborhood $U_0$ of $p_0$ in $X$. For any $p\\in X\\backslash \\partial X$, choose a path $x(t),0\\leq t\\leq 1$ such that $x(0)=p_0, x(1)=p, x(t)\\in X\\backslash\\partial X$, then an analytic continuation of the isometry $\\phi_1|_{U_0}$ along $x(t)$ gives rise to an isometry defined near $p$. We claim that if we have two paths and two analytic continuations, then they define the same germ at $p$. In fact, one only needs to show that for the closed path $y_0(t)$ formed by concatenating these two paths, the isometry near $p_0$ given by an analytic continuation of $\\phi_1|_{U_0}$  along $ y_0(t)$ has the same germ as $\\phi_1|_{U_0}$ at $p_0$. Since $\\pi_1(X,\\partial X)=0$, $y_0(t)$ can be homotoped to a path $y_1(t)$ contained in $U$ via paths $y_s(t)$ in $X\\backslash\\partial X$, such that $y_s(0)=y_s(1)=p_0$. Since $\\phi_1:U\\rightarrow V$ is a globally defined isometry, by uniqueness of analytic continuation, we know that any analytic continuation of $\\phi_1|_{U_0}$  along $y_1(t)$ must coincide with $\\phi_1$, which finishes the proof of the claim by invariance of analytic continuation via homotopy. This shows that $\\phi_1:U\\rightarrow V$ can be extended to a global isometry $\\phi:X\\rightarrow X$ by analytic continuation. Hence $\\bm\\omega_1=\\phi^*\\bm\\omega_2$ by the same argument as before.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nGiven the compactness result,  Theorem \\ref{unique continuation hyperkahler}, together with the theorem of Ebin-Palais on   properness of diffeomorphism group action on the space of Riemannian metrics on a closed manifold (See \\cite{ebin1970manifold}), one can answer the question raised at the beginning of this subsection, thus finish the proof of Theorem \\ref{convergence of triples}, Theorem \\ref{convergence of triples enhancements}:\n \n\n\n\nSuppose we have two sequences of diffeomorphism $\\phi_i,\\psi_i$ of $X$ such that  $\\phi_i^*\\bm\\omega_i\\rightarrow \\bm\\omega$, $\\psi_i^*\\bm\\omega_i\\rightarrow\\bm\\omega'$, then $(\\phi_i|_{\\partial X})^*\\bm\\gamma_i\\rightarrow \\bm\\omega|_{\\partial X}$, $(\\psi_i|_{\\partial X})^*\\bm\\gamma_i\\rightarrow \\bm\\omega'|_{\\partial X}$, where $\\bm\\gamma_i=\\bm\\omega_i|_{\\partial X}$. Since $\\bm\\gamma_i$ converges to   $\\bm\\gamma$ in Cheeger-Gromov sense, there exists diffeomorphisms $u_i:\\partial X\\rightarrow\\partial X$ such that $u_i^*\\bm\\gamma_i\\rightarrow\\bm\\gamma$. By \nthe theorem of Ebin-Palais, we have for a subsequence $(\\phi_i|_{\\partial X})^{-1}\\circ u_i , (\\psi_i|_{\\partial X})^{-1}\\circ u_i $ converge to some diffeomorphisms $u,u'$ on $\\partial X$, respectively(because their inverses converge). \nHence we also have $u_i^*\\bm\\gamma_i\\rightarrow u^*\\bm\\omega|_{\\partial X}$, $u_i^*\\bm\\gamma_i\\rightarrow (u')^*\\bm\\omega'|_{\\partial X}$, so $\\bm\\omega|_{\\partial X}=(u'\\circ u^{-1})^*\\bm\\omega'|_{\\partial X}$. Note that the positive mean curvature condition implies that $\\pi_1(X,\\partial X)=0$ (See Proposition \\ref{diam}),  then by Theorem \\ref{unique continuation hyperkahler}, there exists a diffeomorphism $\\varphi$ on $X$ with $\\bm\\omega'=\\varphi ^*\\bm\\omega$. \n\n\n\n\n\n\\subsection{Some discussions}\n\n\n\n\n\n\nSuppose $X$ is a connected complete metric space, and there exists a finite set $\\Sigma=\\{p_1,\\cdots,p_m\\}, m\\geq 0$ such that $X\\backslash\\Sigma$ is a smooth flat hyperk\\\"ahler\n4-manifold with nonempty boundary, and each boundary component $Y_1,\\cdots, Y_n, n\\geq 1$ is isometric to flat $\\mathbb{R}^3$,  $X\\backslash \\cup_{i=1}^n{Y_i}$ is a flat hyperk\\\"ahler 4-orbifold and $\\Sigma$ is the set of all orbifold points. Our goal is to classify all such $X$.\n\nThe motivation of this problem is the following:\n\\begin{proposition}\\label{classification motivation}\nLet $(X_i,g_i)$ be a sequence of compact hyperk\\\"ahler 4-manifolds with boundary, such that $$i_{b,g_i}\\geq i_0, inj_{\\partial X_i}\\rightarrow\\infty,$$ $$|S_i|\\rightarrow 0, |\\nabla^{j+1}_{\\partial X_i} H_i|\\rightarrow 0, |\\nabla^j_{\\partial X_i} Rm_{\\partial X_i}|\\rightarrow 0$$ uniformly on $\\partial X_i$ for all $j\\geq 0$, and $$\\int_{X_i}|Rm_{g_i}|^2\\leq C.$$ Then for any $p_i\\in X_i$ $d_{g_i}(p_i,\\partial X_i)\\leq K$, a subsequence of $(X_i,g_i,p_i)$ converges in pointed Gromov Hausdorff sense to a complete metric space $(X_\\infty,d_\\infty,p_\\infty)$. The limit $(X_\\infty,d_\\infty)$ satisfies all properties of  $X$ above.\n\\end{proposition}\n\n\nBy Theorem \\ref{good boundary}, the geometry near boundary is nicely controlled,  so this theorem is a consequence of \\cite{anderson1989ricci} or \\cite{bando1989construction} etc.\n\n\\begin{remark}\nIn Theorem \\ref{main theorem hyperkahler metric}, under the mean positive condition, if we allow $-2$ curves in $X_i$, then we are unable to show that $i_{b,g_i}$ has a lower bound. In fact, the bad case is that focal points are exactly those curvature blow-up points, and they approach the boundary in a moderate rate. However, we can say something within the scale of the boundary injectivity radius. Suppose $i_{b,g_i}\\rightarrow 0$, then we rescale the metric such that they are equal to 1, then the rescaled metric satisfies the assumption of Proposition \\ref{classification motivation}.\n\\end{remark}\n\n\n\\begin{theorem}\\label{classification}\n$X$ is isometric to one of the following:\n\\begin{itemize}\n    \\item  $\\text{(m=0, n=1)}$ $\\mathbb{R}^4_+ $;\n    \\item  $\\text{(m=0, n=2)}$   the region in $\\mathbb{R}^4$ bounded by two parallel hyperplanes;\n    \\item $\\text{(m=1, n=1)}$   the connected component of $0$ in   $(\\mathbb{R}^4\\backslash H)\/{\\mathbb{Z}_2}$, where $H$ is a hyperplane such that $0\\notin H$. \n\\end{itemize}\n\n\n\n\\end{theorem}\n\n\n\\begin{proof}\n\nConsider another copy of $X$, glue them together along $Y_1,\\cdots, Y_n$,  we get a complete flat hyperk\\\"ahler orbifold $\\hat{X}=X\\sqcup_{id} X, id:\\cup_{i=1}^nY_i\\rightarrow \\cup_{i=1}^n Y_i\\subset X$, which contains $Y_1,\\cdots, Y_n$ as smooth hypersurfaces. Then $\\hat X$ is a $(SU(2),\\mathbb{R}^4)$ orbifold in the sense of \\cite{thurston1979geometry}. Let  \n$\\tilde{X}$ be the universal covering orbifold of $\\hat X$, then we have a developing map $D:\\tilde {X}\\rightarrow \\mathbb{R}^4$, and since $\\tilde{X}$ is a complete orbifold, $D$ is a covering map.  Hence $\\tilde{X}$ is homoeomorphic to $\\mathbb{R}^4$, and $\\hat X$ is isometric to $\\mathbb{R}^4\/\\Gamma$, where $\\Gamma$ is a discrete subgroup of $\\mathbb{R}^4\\rtimes SU(2)$. Let $r$ be the projection map to the second factor, then $r(\\Gamma)$ is a finite subgroup of $SU(2)$, and we have a short exact sequence\n$$0\\rightarrow \\Gamma\\cap\\mathbb{R}^4\\rightarrow \\Gamma\\rightarrow r(\\Gamma)\\rightarrow 0.$$ \nThen $\\text{rank} (\\Gamma\\cap\\mathbb{R}^4)\\leq 1$, since otherwise $\\hat{X}$ is a quotient of $\\mathbb{R}^2\\times T^2$ and cannot contain a flat $\\mathbb{R}^3$ as a Riemannian submanifold. \n\nIf $\\Gamma\\cap\\mathbb{R}^4=0$, then $\\hat{X}\\cong \\mathbb{R}^4\/r(\\Gamma)$ and\n hence $r(\\Gamma)=\\{1\\}$, since otherwise $\\hat{X}$ has exactly one orbifold point, contradiction. Hence, $m=0, n=1$ and  ${X}$ is isometric to $\\mathbb{R}_+^4$.\n\nIf $\\Gamma\\cap\\mathbb{R}^4=\\mathbb{Z}a$ for some $0\\neq a\\in\\mathbb{R}^4$, let $\\pi: \\mathbb{R}^4\\rightarrow \\mathbb{R}^4\/\\Gamma$ be the covering map, then $\\pi^{-1}(Y_1)$  is a complete totally geodesic submanifold of $\\mathbb{R}^4$, hence a countable disjoint union of parallel hyperplanes.\nPick one of them $Z$, then $\\pi^{-1}(Y_1)$ is a disjoint union of $\\gamma. Z$ for $\\gamma\\in \\Gamma$. Suppose $Z$ is defined by $b^Tx+c=0$, then $\\gamma^{-1}.Z$ is defined by $b^T r(\\gamma)x+c'=0$. Since they are parallel to each other, $b^T=\\pm b^Tr(\\gamma)$, and hence $\\pm 1$ is an eigenvalue of $r(\\gamma)$, which forces $r(\\gamma)=\\pm 1$. Hence $r(\\Gamma)=\\{1\\}$ or $r(\\Gamma)=\\mathbb{Z}_2$. If $r(\\Gamma)=\\{1\\}$, then $\\Gamma$ is generated by $x\\mapsto x+a$, so $m=0, n=2$ and $X$ is isometric to the region in $\\mathbb{R}^4$ bounded by two parallel hyperplanes; If $r(\\Gamma)=\\mathbb{Z}_2$, then $\\Gamma$ is generated by $x\\mapsto -x+d$ and $x\\mapsto x+a$ for some $d\\in\\mathbb{R}$. Hence $m=1$ and $\\mathbb{R}^4\/\\Gamma\\backslash\\{Y_1,Y_2,\\cdots, Y_n\\}$ has $n+1$ connected components. But from the gluing construction and that $X$ is connected, we know $\\hat{X}\\backslash\\{Y_1,Y_2,\\cdots, Y_n\\}$ has exactly two connected components. Hence $n+1=2, n=1$, and $X$ is isometric to the last case in the conclusion.\n\n\n\\end{proof}\n\n\n\\section{Torsion-free hypersymplectic manifolds with boundary}\\label{section torsion-free triples}\n\n\\subsection{Preliminaries}\n\nRecall that a hypersymplectic triple $\\bm\\omega=(\\omega_1,\\omega_2,\\omega_3)$ on an oriented 4-manifold $X$ with boundary is a definite triple of symplectic forms. Write\n $$\\omega_i\\wedge\\omega_j=2Q_{ij}\\mu$$\n Denote $\\bm{Q}=(Q_{ij})$, $\\bm{Q}^{-1}=(Q_{ij})$, and $g=g_{\\bm\\omega}$ the Riemannian metric. Recall that $\\bm\\omega$ is called torsion-free \nif for each $i$,\n \\begin{equation}\n      d(Q^{ij}\\omega_j)=0,\n \\end{equation} which is equivalent to\n\\begin{equation}\\label{torsion-free condition}\n    dQ^{ij}\\wedge \\omega_j=0.\n\\end{equation}\n\nLet us begin with some arguments and results in \\cite{fine2020report}. Let $\\mathcal{P}$ denotes the set of symmetric positive-definite 3 by 3 matrices. There are two Riemannian metrics on $\\mathcal{P}$: the first one is the Euclidean metric \n\n\\begin{equation}\\label{usual laplacian torsion-free}\n    \\langle A, B\\rangle =Tr(AB),\n\\end{equation}\n and the second one is the symmetric space metric, which has non-positive sectional curvature\n\\begin{equation}\\label{hat laplacian torsion-free}\n     \\langle A, B\\rangle_Q=Tr(Q^{-1}A Q^{-1}B) \n\\end{equation}\nat each point $Q\\in \\mathcal{P}$.\n\nThen $\\bm{Q}$ can be regarded as a map $\\bm{Q}:X\\rightarrow \\mathcal{P}$. Let $\\Delta \\bm Q, \\hat{\\Delta}\\bm Q$ denote the harmonic Laplacian of this map with respect to (\\ref{usual laplacian torsion-free}),(\\ref{hat laplacian torsion-free}), respectively. Explicitly, their components are related by\n\\begin{equation}\\label{torsion-free Laplacian operator}\n    (\\hat\\Delta\\bm Q)_{ij}=\\Delta Q_{ij}-Q^{km}\\langle dQ_{ik},dQ_{mj}   \\rangle,\n\\end{equation}\n\n\nFor a hypersymplectic triple $\\bm\\omega$ , the calculations in \\cite{fine2018hypersymplectic} showed that the torsion-free condition is equivalent to \n\\begin{equation}\\label{torsion-free equations}\n\\hat\\Delta\\bm Q=0, \\ \\text{Ric}= \\frac{1}{4}\\langle d\\bm Q\\otimes d\\bm Q\\rangle_{\\bm Q},\n\\end{equation}\nwhere $\\langle d\\bm Q\\otimes d\\bm Q \\rangle_{\\bm Q}(u,v)=\\langle \\nabla_u \\bm Q,\\nabla_v \\bm Q\\rangle_{\\bm Q}$.\nHence if $\\bm\\omega$ is torsion-free, then $\\bm Q$ is a harmonic map with respect to (\\ref{hat laplacian torsion-free}) and  $\\text{Ric}\\geq 0$. Then the scalar curvature $R$ of $g$ is $$R=\\frac{1}{4}|d\\bm Q|_{\\bm Q}^2\\geq 0,$$ \nwhich is a multiple of the energy density of the harmonic map $\\bm Q$.\nTake the trace of (\\ref{torsion-free Laplacian operator}), we get \n\\begin{equation}\\label{subharmonic TrQ}\n     \\Delta Tr \\bm Q=Q^{pq}\\langle dQ_{kp},dQ_{qk}\\rangle\\geq 0.\n\\end{equation}\nMoreover, \\cite{fine2020report} showed that the function $R$ satisfies the inequality \n\\begin{equation}\\label{R Delta R}\nR\\Delta R\\geq \\frac{1}{2}|\\nabla R|^2+\\frac{1}{2}R^3,\n\\end{equation}\nhence a contradiction argument implies that everywhere\n\\begin{equation}\\label{subharmonic scalar curvature}\n    \\Delta R\\geq 0.\n\\end{equation}\nThen they used standard geometric analysis arguments for inequality (\\ref{R Delta R}) and the fact $\\text{Ric}\\geq 0$ to conclude\n\n\\begin{proposition}\\label{fy result}\nSuppose $B(p,r)\\subset X$ has compact closure, and $\\partial B(p,r)\\neq \\emptyset$, then \n$R(p)\\leq \\frac{32}{r^2}$. In particular, a complete torsion-free hypersymplectic 4-manifold is hyperk\\\"ahler.\n\\end{proposition}\nNote that in the latter case, there exists a constant matrix $B$ in $SL(3,\\mathbb{R})$ such that $\\bm \\omega B$ is a hyperk\\\"ahler triple, and the Riemannian metric defined by  $\\bm \\omega$ is the same as the one defined by $\\bm\\omega B$. \n\n\\subsection{Compactness for the boundary value problem}\n \nNow suppose $X$ is an oriented 4-manifold with compact boundary $Y=\\partial X$, then through the boundary exponential map, a neighborhood $U$ of $Y$ is diffeomorphic to $Y\\times [0,a)$. Let $t$ denote the distance function $d(\\cdot, Y)$, i.e., the projection $Y\\times [0,a)\\rightarrow [0,a)$, then in $U$, $\\bm\\omega$ can be written as\n $$\\bm\\omega=-dt\\wedge*_{Y_t}\\bm\\gamma_t+\\bm\\gamma_t.$$\n where $Y_t=Y\\times \\{t\\}$ and $*_{Y_t}$ is the Hodge star operator of $g|_{Y_t}$.\n $\\bm\\omega$ being closed is equivalent to \n \\begin{equation}\n     d_{Y_t}\\bm\\gamma_t=0,\n \\end{equation}\n \\begin{equation}\\label{torsion-free normal tangential 0}\n     \\frac{\\partial \\bm\\gamma_t}{\\partial t}=-d_{Y_t}(*_{Y_t} \\bm\\gamma_t).\n \\end{equation}\n  (\\ref{torsion-free condition})\n is equivalent to \n\\begin{equation}\\label{torsion-free normal tangential}\n    d_{Y_t}Q^{ij}\\wedge *_{Y_t}\\gamma_{t,j}+\\frac{\\partial Q^{ij}}{\\partial t}\\gamma_{t,j}=0,\n\\end{equation}\n \\begin{equation}\n     d_{Y_t}Q^{ij}\\wedge \\gamma_{t,j}=0.\n \\end{equation}\n Note that\n (\\ref{torsion-free normal tangential}) is equivalent to\n \\begin{equation}\\label{torsion-free normal tangential 2}\n     \\frac{\\partial Q^{ij}}{\\partial t}=\\frac{d_{Y_t}Q^{ik}\\wedge \\eta_{t,j}\\wedge *_{Y_t}\\gamma_k}{\\text{vol}_{t}},\n \\end{equation}\n where $\\text{vol}_t=\\eta_{t,1}\\wedge \\eta_{t,2}\\wedge\\eta_{t,3}$ and equals to the Riemannian volume form of $g|_{Y_t}$.  From calculations in Lemma \\ref{hyperkahler triple second fundamental form} and (\\ref{torsion-free normal tangential 0}), (\\ref{torsion-free normal tangential 2}) , it is easy to see that the second fundamental form $II(e_i,e_j)$ is in algebraic terms of $\\bm\\eta, d_{\\partial X}\\bm\\eta,\\bm Q, d_{\\partial X}\\bm Q$.\n\n \n\n Now let us try to prove Theorem \\ref{convergence of hypersymplectic triples}, starting with basic observations.\n In Theorem \\ref{convergence of hypersymplectic triples}, suppose $\\bm\\omega_i|_{\\partial X}, \\bm Q_i$ converge in Cheeger-Gromov sense to the limit,\n then we have $\\text{diam}(\\partial X,g_i|_{\\partial X})\\leq C, \\text{vol}_{g_i}(\\partial X)\\leq C, inj_{\\partial X, g_i|_{\\partial X}}\\geq i_0$,  $|\\nabla_{\\partial X}^j Rm_{\\partial X,g_i}|\\leq C_j$, $|\\nabla ^j_{\\partial X} S_i|\\leq C_j$. By (\\ref{subharmonic scalar curvature}) and maximum principle, $R_i$  is uniformly bounded on $X$, so $|\\text{Ric}_{g_i}|$ is uniformly bounded on $X$ since $\\text{Ric}_{g_i}$ is non-negative and its trace is uniformly bounded.  Due to the uniform mean positive curvature condition, and $\\text{Ric}_{ g_i}\\geq 0$,  we have an upper bound of $\\sup\\limits_{p\\in X}d_{g_i}(p,\\partial X)$, $\\text{vol}_{g_i}(X)$ by Proposition \\ref{diam}, and in particular an upper bound of $\\text{diam}(X, g_i)$. The Chern-Gauss-Bonnet formula thus gives an upper bound of $\\int_{X}|Rm_{g_i}|^2$. Also, by (\\ref{subharmonic TrQ}) and maximum principle, $Tr \\bm Q_i$ is uniformly bounded on $X$, so $\\bm Q_i$ is uniformly bounded on $X$ and then $\n|d\\bm Q_i|$ is uniformly bounded on $X$. \n\n Given these conclusions, we need to verify that all propositions that were used to prove Theorem \\ref{convergence of triples enhancements} adapt to the torsion-free hypersymplectic setting. Firstly, we digress to discuss elliptic regularity for torsion-free equations (\\ref{torsion-free equations}). In harmonic coordinates in $B_2$ or $B_2^+$, view $g$ as a 4 by 4 matrix of functions, then we have a system of PDEs in $g, Q$: \n \\begin{equation}\\label{torsion-free equation 1 harmonic}\n     \\Delta_g Q_{kl}-Q^{pq}\\langle dQ_{kp},dQ_{ql}\\rangle_g=0,\n \\end{equation}\n \\begin{equation}\\label{torsion-free equation 2 harmonic}\n     \\Delta_g g_{ij}+B_{ij}(g,\\partial g)=-\\frac{1}{2}Q^{ab}Q^{cd}{\\partial_i}Q_{bc}{\\partial_j}Q_{da}.\n \\end{equation}\n From this, by a boostrapping argument, one sees interior regularity: fix $\\beta\\in (0,1)$. If $\\bm Q$ is $C^1$ bounded, $g$ is $C^{1,\\beta}$ bounded, and they are uniform positive on $B_2$, then all derivatives of $\\bm Q$ and $g$ are bounded. For boundary regularity, there is no techinical difficulty to get the following version from Neumann boundary conditions (\\ref{Neumann boundary condition 1}),(\\ref{Neumann boundary condition 2}): if all tangential\n derivatives of $\\bm Q$, $g$, $H$ are bounded on $\\tilde{B}_2$, and  $\\bm Q$ is $C^1$ bounded, $g$ is $C^{1,\\beta}$ bounded, and they are uniformly positive on $B_2^+$, then all derivatives of $\\bm Q$ and $g$ are bounded in $B_1^+$. If in both cases, we also assume $g$ is $C^{k,\\beta}$ close to identity in $B_2$ or $B_2^+$, then similarly, $g$ is $C^{k,\\alpha}$ close to identity for any $\\alpha\\in (\\beta,1)$.\n\n\nFollowing the proof of Theorem \\ref{local harmonic radius lower bound}, we get the following two propositions.\n\n\\begin{proposition}\\label{harmonic radius and bounds for Q} Let $(X,g,\\bm\\omega)$ be a  torsion-free hypersymplectic manifold and $B(p,r)$ is a metric ball that has compact closure, $\\partial B(p,r)\\neq \\emptyset$. Suppose  for any $q\\in B(p,r)$, $$inj_q \\geq cd(q,\\partial B(p,r)), $$\n$$Tr\\bm Q, R\\leq  C.$$\nFix $\\Lambda>1,0<\\alpha<1$, then for any $k\\geq 0$,  $q\\in B(p,r),$  $$r_h^{k,\\alpha}(q,g, \\Lambda)\\geq C_k'd(q,\\partial B(p,r)).$$\nIn particular, $|\\nabla^k \\bm Q|\\leq C_k''$ in $B(p,\\frac{r}{2})$.\n\n\\end{proposition}\n\n\n\\begin{proposition} Let $(X,g,\\bm\\omega)$ be a  compact torsion-free hypersymplectic manifold with boundary. Suppose $i_b\\geq i_0$, $inj_{X}\\geq i_0$, $inj_{\\partial X}\\geq i_0$, $Tr \\bm Q$, $R\\leq C$ on $\\partial X$ and $|\\nabla^j_{\\partial X} Rm_{\\partial X}|\\leq C_j$, $|\\nabla^j_{\\partial X} S|\\leq C_j $, $|\\nabla^j_{\\partial X} \\bm Q|\\leq C_j$ on $\\partial X$, $\\forall k\\geq 0$. Fix $\\Lambda>1, 0<\\alpha<1$, then for any $k\\geq 0$, $q \\in X$,\n$$r_h^{k,\\alpha}(q,g, \\Lambda)\\geq C_k'''.$$\n\\end{proposition}\n \n Then we have \n \n\\begin{proposition}\nProposition \\ref{no bubble 2} holds for torsion-free hypersymplectic manifolds $(X,g,\\bm\\omega)$, provided an upper bound of $Tr\\bm Q$ in $B(p,5)$.\n\\end{proposition}\n\n\\begin{proof}\n\nThe proof is almost the same as there. Let us list the ingredients here: \n\n\\begin{itemize}\n\\item We have Bishop-Gromov volume comparison, since $\\text{Ric}_{g_i}\\geq 0,$\t\n\\item Before rescaling, $ R_i, |\\text{Ric}_{g_i}|$ are automatically bounded by Proposition \\ref{fy result}.\n\\item \n  For the rescaled metric $\\tilde{g}_i$, the \n  curvature bound and the volume non-collapsing condition imply injectivity radius lower bound on compact sets, hence by Proposition \\ref{harmonic radius and bounds for Q}, we have harmonic radius lower bounds as well as bounds for derivatives of $\\bm Q_i$ on compact sets, so we have pointed Cheeger-Gromov convergence of a subsequence $(M_i,g_i,\\bm\\omega_i, q_i)$.\n\n\\item The limit $\\bm\\omega_\\infty$ is a hyperk\\\"ahler triple up to a $SL(3,\\mathbb{R})$ rotation, because $g_\\infty$ is scalar flat, or because of Proposition \\ref{fy result}.\n\n\\end{itemize}\nSo, we get the contradiction in the same way. \n\\end{proof}\n\nWith the above three Propositions as tools and $\\epsilon$-regularity, argue the same way as in Theorem \\ref{good boundary}, one gets an analogous version of Theorem \\ref{good boundary}, i.e., curvature control within  $i_b$, assuming $i_b\\geq i_0$.  \n\n\\begin{remark}\nWe make a remark about the proof of $\\epsilon$-regularity for torsion-free hypersymplectic manifolds here. Firstly, the proof in \\cite{sun2021collapsing} Theorem 3.21 directly applies to this case by using Proposition \\ref{fy result}. Alternatively, we can apply Remark 8.22 in \\cite{cheeger2006curvature} to conclude that $g$ has $C^{1,\\alpha}$ bounded covering geometry when curvature $L^2$ norm is small. By (\\ref{torsion-free equations}), we have $|\\nabla \\text{Ric}|\\leq C$, hence $g$ has $C^{2,\\alpha}$ bounded covering geometry and $|Rm|$ is bounded.\n\\end{remark}\n\n\n \nNow one can finish the proof of compactness part of  Theorem \\ref{convergence of hypersymplectic triples} by the same arguments in Section \\ref{section main proof}. Note that $\\text{Ric}\\geq 0, H>0$ is enough for the focal point argument.\n\n\\subsection{Uniqueness}\n\nFinally, we prove the uniqueness part of Theorem \\ref{convergence of hypersymplectic triples}.\n\n\\begin{proposition} Let $\\bm\\omega_1,\\bm\\omega_2$ be two torsion-free hypersymplectic triples on an oriented 4-manifold $X$ with compact boundary $Y=\\partial X\n$.\nSuppose $\\bm\\gamma_1=\\bm\\gamma_2, \\bm Q_1=\\bm Q_2$ on $\\partial X$, then $\\bm\\omega_1=\\bm\\omega_2$ in geodesic gauges of $g_{\\bm\\omega_i}$ near $\\partial X$.\n\\end{proposition}\n\n\\begin{proof}\n$\\bm\\omega_i$ defines a torsion-free $G_2$ structure $\\phi_i$ on $X\\times T^3$ via (\\ref{g2 and hypersymplectic}), which defines a warpped product metric \n\\begin{equation}\\label{warpped product metric}\n    g_{\\phi_i}=g_{\\bm\\omega_i}+ Q_{ij}dt^idt^j.\n\\end{equation}\nIn the geodesic gauge of $g_{\\bm\\omega_i}$, write\n$$\\bm\\omega_i=-dt\\wedge *_{Y_t}\\bm\\gamma_t+\\bm\\gamma_t,$$\nwhere $t$ is the distance function $d_{g_{\\bm\\omega_i}}(\\cdot,\\partial X)$.\nHence \n\\begin{equation}\\label{g2 geodesic gauge}\n\\phi_i=-dt\\wedge \\theta_{t,i}+\\rho_{t,i},\n\\end{equation}\nwhere $$\\rho_{t,i}=dt^1\\wedge dt^2\\wedge dt^3-\\gamma_i^1\\wedge dt^1-\\gamma_i^2\\wedge dt^2-\\gamma_i^3\\wedge dt^3,$$\n$$\\theta_{t,i}=-*_{Y_t}\\gamma_{t,i}^1\\wedge dt^1-*_{Y_t}\\gamma_{t,i}^2\\wedge dt^2-*_{Y_t}\\gamma_{t,i}^3\\wedge dt^3$$\nBy (\\ref{warpped product metric}), $t$ can also be viewed as $d_{g_{\\phi_i}}(\\cdot,\\partial X\\times T^3)$, so (\\ref{g2 geodesic gauge}) is written in the geodesic gauge of $g_{\\phi_i}$.\nBy the calculations in \\cite{donaldson2018remarks} Section 2.2, for $i=1,2$, both $g_{\\phi_i}|_{\\partial X\\times T^3}$ and the second fundamental forms of $\\partial X\\times T^3$ are equal to each other, since they are explicitly in terms of $\\theta_{0.i},\\rho_{0,i}$, which are in terms of $\\bm\\gamma_i,\\bm Q_i$. Since $g_{\\phi_i}$ are Ricci-flat, by \\cite{biquard:hal-02928859} Theorem 4, $g_{\\phi_1}=g_{\\phi_2}$. Since $\\nabla^{g_{\\phi_1}}|\\phi_1-\\phi_2|^2=0$ and $\\phi_1-\\phi_2=0$ at one point, we have $\\phi_1=\\phi_2$, $\\bm\\omega_1=\\bm\\omega_2$.\n\n\\end{proof}\n\nNote that for a torsion-free hypersymplectic triple $\\bm\\omega$, the metric $g_{\\bm\\omega}$ is real analytic with respect to the analytic structure defined by harmonic coordinates, due to elliptic regularity of (\\ref{torsion-free equation 1 harmonic})(\\ref{torsion-free equation 2 harmonic}), so the arguments in subsection \\ref{uniqueness 1} shows global uniqueness:\n\n\n\\begin{theorem}\\label{unique continuation torsion-free hypersymplectic}\nLet $X$ be a connected 4-manifold with boundary, $\\pi_1(X,\\partial X)=0$. Suppose $\\bm\\omega_1,\\bm\\omega_2$ are two smooth torsion-free hypersymplectic triples on $X$, and $\\varphi_0:\\partial X\\rightarrow\\partial X$ is a diffeomorphism, such that $\\bm\\omega_1|_{\\partial X}=\\varphi_0^*(\\bm\\omega_2|_{\\partial X})$, $\\bm Q_1|_{\\partial X}=\\varphi_0^*\\bm Q_2|_{\\partial X}$, then there exists a diffeomorphism $\\varphi:X\\rightarrow X$,   $\\varphi|_{\\partial X}=\\varphi_0$,  such that $\\bm\\omega_1=\\varphi^*\\bm\\omega_2$ on $X$.\n\\end{theorem}\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}